The Original Beatle Cassettes (1968-69)
The Dutch electronics company Phillips introduced the Compact Cassette Tape in 1963 but it wasn't until 1968 that the first Beatles album was made available on cassette in the UK, Sgt.Pepper's Lonely Hearts Club Band. The original Pepper cassettes were issued with green labels and included a distinctive Parlophone logo, the first pressings included a "Sold in the UK..." notice in the bottom right hand corner and the leader tape was stamped with the catalogue number, see examples here and here. In 1970 the labels were changed to yellow and by 1971 they were being stamped with a new black-boxed style Parlophone design incorporating EMI's logo above it. Abbey Road was the 2nd Beatles album to appear on cassette and this also had green labels when it was first issued in late 1969, but there was no Apple or Parlophone logo.
Unlike the 1970's inlays, the original designs for these two albums did include some extra artwork other than just a reproduction of the front cover, the Pepper inlay foldover included black & white cut outs of the Beatles heads (as originally posed for the vinyl gatefold) whilst the Abbey Road foldover had a colour reproduction of the back cover of the vinyl jacket.
As with all Beatles cassettes issued until 1987, the cassette boxes (or to use the correct term, the Norelco cases) were clear plastic at the front and around the spine area (to reveal the inlay cover) and black plastic at the rear. The actual cassette shells were usually white/light grey (some may have been issued in black).
 

White Inlays (1970-72)
Let It Be was the 3rd Beatles album to be issued on cassette and this inlay design was to be the template for the release of the Beatles back catalogue. Rubber Soul, Revolver and Oldies But Goldies were first issued in September 1970 followed by Please Please Me,With The Beatles, A Hard Day's Night, Beatles For Sale and Help! in October 1970 and finally The White Album in mid-1971*. This left only Yellow Submarine unissued on cassette. For the re-arranged running orders, see here.
The 1970/71 cassettes all had white inlays and, aside from the small reproduction of the front cover, no artwork was included nor the original liner notes. The tracklistings were printed on the reverse of the inlay whilst the foldovers offered a list of other available Beatles/solo cassettes. The two exceptions to this rule were Revolver - which had a black spine and a black top above the front cover artwork - and The White Album, originally issued inside a cigarrette type packet which slipped into the back of a large display card. The White Album was only available in 'display card' form for a short period before being replaced by a standard white inlay design although this particular version didn't promote any other releases.
The cassette shell labels during this period were yellow. Original prints of Let it be carried the old style Parlophone logo whilst the other albums either had the new style black-boxed Parlophone/EMI logo or no logo at all.
All Beatle cassettes issued by EMI before 1987 were stereo including the first four albums which would later only be available in mono on CD and 'XDR' cassette from 1987-2009. At the time these original versions were issued, cassettes were only just beginning to challenge the long established Reel to Reel tape fromat to become the second most popular method of listening to music behind vinyl, so I doubt many original cassettes were manufactured or sold. Originals in good condition should now command a reasonable price and in time could become highly collectable.

*The publishing credits on the early White Album cassettes suggest that it was originally issued in the Spring/Summer of 1969, but the IE number points to mid-1971
Sgt. Pepper and Abbey Road continued to be sold with their slightly more elaborate inlays which included extra artwork other than just the album cover.
 

All Gold Inlays (1972-73) [re-issued with subtle variations until the mid-1980's]
All of the regular UK Beatle albums (plus Oldies But Goldies but excluding Yellow Submarine) began to appear in the shops with all-gold inlays in 1972. As with the earlier white inlays, there was no artwork except for the front cover*. The tracklistings were now printed on the gold coloured foldover whilst the reverse of the inlay was blank except for a copyright/printing notice and the assurance that "This stereo 'Musicassette' also gives genuine monaural reproduction." The White Album and Let it be gold inlays both had the Apple logo added to the backflip and the legend "An Apple Record" printed on the foldover, but for some reason neither of these additions appeared on the Abbey Road inlay.
Magical Mystery Tour joined the Beatles cassette catalogue in June 1973, this being the first British release of the American compilation although it evidently used the German master as it contained true stereo mixes of "Penny Lane", "Baby You're a Rich Man" and "All You Need is Love". The British Parlophone vinyl edition of Magical Mystery Tour was not issued until 1976 (and this used the U.S. master and therefore contained reprocessed mono mixes of the aforementioned songs). The 1973 'Red' and 'Blue' compilations were also issued on cassette shortly before the next change of inlay design as they too had the all-gold look.
Gold variations
The all-gold inlays were originally accompanied by the same old yellow labelled cassette shells, some being stuck on to black shells rather than the more familiar light grey**. In 1973 Dolby noise reduction was introduced to contemporary solo cassettes which appears to have co-incided with the introduction of white labels, but the regular Beatles albums were not manufactured with Dolby until later in the decade, by which time labels had been dropped in favour of printing the details directly onto the cassette shell (including a Dolby logo) but the inlay design remained unchanged***. The ink colour on the Dolby shells varied, bright blue (circa 1977-82), brown (1982/83), pink (1983-85) and finally black. All directly printed ink versions included a Parlophone/EMI or silhouetted Apple logo.
There are also different shades of gold inlays in circulation, the earlier prints have a darker metallic look to them which reveal fingerprints and commonly suffer from surface erosion to the foldover, caused by the cassette rubbing against the gold paint when taken in and out of the case, later versions (probably first seen around 1976) are a lighter yellowy-brown although the 'Red' and 'Blue' compilation albums continued to be issued with metallic gold inlays until they were re-issued in 1993.
Mid 1980's pressings of the gold cassettes added a bar code to the back flip and the IE number was dropped from the spine. From 1986/87 the tape shells were a creamy yellow.

* In the early 1970's, due to some sort of dispute, the Parlophone logo was airbrushed from the original "With The Beatles" and "Help!" cover designs for reproduction on the gold cassette inlays.
** In comparison to the original late 1960's green labels, the early-mid '70s labels do appear to be a bona-fide yellow but some may actually be chartreuse.
*** I have yet to see any white labelled regular Beatle album cassettes with a dolby logo - this would suggest that Dolby versions of the Beatles albums first appeared in late 1976 or shortly afterwards.
All gold & 'Circular EMI' inlay designs include the legend "Also available on 8 track cartridge", this early-mid 1970's rival to cassette was easily beaten off - particularly in Britain - and became obsolete by the end of the decade. Cassettes would go on to outsell vinyl from 1982.
 

Gold Top with Circular EMI Logo (1973-1976)
Yellow Submarine was the last of the regular British albums to be released on cassette in August 1974 (likely prompted by the British television premiere of the film in April) and this had a cassette inlay design first seen with EMI tapes from mid-late 1973, a gold band across the top third of the inlay and the contemporary circular style EMI logo in the top right corner. The Dolby noise reduction logo also made an appearance on the front as well as the spine and there was a paragraph of advise about Dolby on the inlay reverse (alongside the customary copyright/mono playback notices). Regrettably, the front cover of the vinyl sleeve was once again the only artwork to be included on these 'Circular EMI' inlays.
Yellow Submarine was the only original Beatle album to be issued in this style, but there were a good number of solo Beatle albums released in this form as well as the 1976 Rock 'n' Roll Music compilation*. Original copies of Yellow Submarine had white labels (the first Beatle cassette labels to carry the Apple logo) as well as metallic gold paint on the inlay.

*Rock 'n' Roll Music was issued as a "double play tape" with a re-arranged tracklisting, the compilation was later split into two separate volumes for budget re-issues on the MFP label in 1980 (these had tracklistings which matched the vinyl versions).
 

'Full Height' Covers (1977-1982)
1976/77 saw the introduction of 'full height' inlay covers which sometimes meant losing the left and right edges of the original artwork (as in the case of the 1979 UK cassette release of the American compilation Hey Jude) or filling up the space above/below with something in harmony with the look of the vinyl cover (as illustrated left by the 1980 compilation The Beatles Ballads).
No standard 1960's Beatles album was issued 'full height' but a number of compilations were. These new designs still had a uniform look - in as much as the lettering on the spine and the basic layout of information was standard across all EMI cassettes. This style continued until the mid 1980's when inlays finally began to expand into full colour multiple-foldover designs that attempted to appear as striking and unique as their increasingly lavish vinyl counterparts.
 
 

Compilations on Cassette 1977-1987
The Beatles At The Hollywood Bowl 1977, Love Songs* 1977, Rarities 1979**, Hey Jude 1979, The Beatles Ballads 1980, Rock 'N' Roll Music Volume 1 1980***, Rock 'N' Roll Music Volume 2 1980***, The Beatles Box 1980****, Reel Music 1982 and 20 Greatest Hits 1982.
* Love Songs was issued as a "double play tape"
**Rarities was also included in The Beatles Collection which was a box set containing all of the Beatles regular albums (excluding Magical Mystery Tour but adding a new 'exclusive' compilation entitled Rarities) the box set was made available in vinyl form on 2nd December 1978, but owing to demand (as well as unscrupulous dealers taking the albums out of the box and selling Rarities for an extortionate price) Rarities was officially made available separately from October 1979 with the cassette version appearing a month later (presumably the cassette box set was also released in November 1979).
***Rock 'N' Roll Volumes 1 & 2 were budget priced "Music for Pleasure" re-issues of the originally combined 1976 double album.
****The Beatles Box was an eight LP/Cassette mail-order compilation, it was made available again in 1987 but this time on cassette only.

Marks & Spencer's "Their Greatest Hits" Compilation
In November 1984, the high street retailer Marks & Spencer released The Beatles: Their Greatest Hits on cassette only, it was part of an exclusive Beatles package that also included a book (mostly featuring material first published in the "Story of Pop" series in 1973) . As with all Marks & Spencer products, the cassette was branded with a St Michael logo and, although the cover was an exclusive design (see here) the contents was a track-for-track replica of Oldies But Goldies (presented here in the same order as the vinyl edition, not the re-arranged order as found on all previously available cassette versions of the 1966 compilation). The set was only on sale at larger branches of M&S and for a limited period of two months, it was then re-promoted the following year (again for a limited period).

The Orbis Cassette
Collectors of the magazine series "The History Of Rock" were offered the chance to collect a series of mail order albums in either vinyl or cassette format, The History Of Rock Volume 26 - The Beatles (released in December 1984) was a 'double play tape' which combined Oldies But Goldies on one side with The Beatles at the Hollywood Bowl on the other, the cover was a minimalist design matching the other volumes which would total 30 in number.

Heineken "Only The Beatles" Compilation
In June 1986, Heineken Beer launched a promotion in which you could order an exclusive Beatles compilation (on cassette only) in exchange for beer can ring pulls. The tape included a previously unreleased stereo version of Yes It Is (later included on Past Masters). Within weeks of the offer first being made, Apple sued and the tapes were withdrawn resulting in Only The Beatles becoming perhaps the most collectable Beatles cassette of all.
 

XDR [eXtended Dynamic Range] Cassettes (1987-)
The "XDR Quality System" tapes were the final generation of UK Beatle cassettes, they were issued during the same year that the Beatles albums were first issued on Compact Disc (1987) and, for the first time, all the cassettes presented the tracks in the correct running order*.
The XDR inlays were mainly white but included all of the original artwork found on the back covers and gatefolds of the vinyl editions. Sgt. Pepper was the exception as this boasted a full height cover and a red foldover/spine as well as a reproduction of the gatefold. The White Album, which had previously been issued as a single "Double Play Tape", was now issued on a pair of cassettes (so like the vinyl version, it was now spread over 4 sides).
The XDR cassette shells were clear plastic with details printed in white ink (including a Parlophone logo) and the cassettes were all housed in completely clear boxes (front AND back). As with the CD's, original XDR tapes (including The White Album through to Let it be) did NOT include any Apple logo's on the inlay or on the shell, it was only from around 1992/93 that re-pressings of Beatles cassettes and CD's began to include the full colour Apple logo's.
By 1993 Compact Discs (which had been introduced to the High Street a decade earlier) were outselling cassettes and with the advent of CD-R and MP3 players at the turn of the Millennium, the cassette era came to an end.

*Incredibly, Capitol in the USA inexplicably used the old re-arranged orders (and gold inlay designs) for the first 7 XDR cassette albums. Perhaps they took instructions to issue ''all future releases in identical fashion to the British versions" too literally.
 

Compilations on Cassette 1988-2003
The twin cassette combination of the Past Masters volumes (released as two separate albums on CD) was delayed to co-incide with the 31st October 1988 release of the XDR cassette box set The Beatles Ultimate Collection, whilst the ever popular red and blue compilation albums had to wait until 1993 for a modern cassette re-issue. The next EMI Beatle release on cassette was Live at the BBC in 1994 followed by the Anthology albums in 1995/96, then Yellow Submarine Songtrack in 1999 and "1"  in 2000. It is not clear whether or not Let it be...Naked (2003) was blessed with a bona-fide British cassette release, but for certain it was issued on cassette in other territories. There was no XDR version of Oldies But Goldies or any of the EMI compilations issued in the late 1970's/early 1980's.

Cassette Singles
The first release of a Beatles single on tape did not occur until 13th July 1987 when EMI added a cassette version of All You Need Is Love/Baby You're A Rich Man to the the list of formats being made available to co-incide with the single's 20th anniversary (EMI had been issuing anniversary picture discs of each Beatles single since 1982, there were a handful of 12" singles too). Both sides of the All You Need Is Love single were recorded onto each side of the short-play cassette (apparently in mock mono!) but the tape did not prove popular enough to warrant EMI issuing further cassette versions of the remaining Beatles singles until 12th August 1991 when all 22 singles were issued in one go, a display case was produced but this was not officially on sale to customers who were expected to purchase them all separately. In keeping with the 3" CD re-issues, the true mono single mixes were used for the cassette singles up to and including Get Back/Don't Let Me Down whilst the covers for these cassettes (housed in slip case boxes rather than Norelco cases) matched the 3" CD Single sleeves (therefore Hey Jude/Revolution had the 'Mad day out' parrot photo rather than the Revolution promo-film shot that was used on the 1980's vinyl picture sleeve). In 1994 the four track BBC Radio sessions EP Baby It's You was released on cassette, Free As A Bird/Christmas Time (Is Here Again) and Real Love/Baby's In Black were issued as two track cassette singles in 1995 & 1996.
 
 

Cassette Scans

Original inlays (1968-1970)

1968 Sgt. Pepper - inlay / inlay reverse / tape
1969 Abbey Road - inlay / inlay reverse / tape
1970 Let it be - inlay / inlay reverse / tape / tape (without logo)

White inlays (1970-1971)

Please Please Me - inlay / inlay reverse / tape
With the Beatles - inlay / inlay reverse / tape
A Hard Day's Night - inlay / inlay reverse / tape
Beatles For Sale - inlay / inlay reverse / tape
Help! - inlay / inlay reverse / tape
Rubber Soul - inlay / inlay reverse / tape
Revolver - inlay / inlay reverse / tape
Oldies But Goldies - inlay / inlay reverse / tape
White Album - inlay

Gold re-issues (1972-1987)
*Early versions (circa 1972-1976) had metallic gold paint.
Bar-coded gold inlays appeared around 1985

Please Please Me
With The Beatles
A Hard Day's Night
Beatles For Sale
Help!
Rubber Soul
Revolver*
Oldies But Goldies
Sgt.Pepper*
White Album
Abbey Road*
Let it be*

Standard gold inlay reverse

Gold issue label examples
*Gold inlay's were originally accompanied by the same old yellow labelled cassettes, but these soon gave way to white labels.
Directly printed blue inked cassettes began to appear around 1976/77 (these being the first Dolby versions of the Beatles albums), a brief run of brown inked versions was followed by pink in the early 1980's.

Oldies But Goldies (white label / grey shell, circa 1973)
Sgt.Pepper (white label / black shell, circa 1974)
Revolver (blue ink, late 1970's)
White Album (pink ink, early/mid 1980's)
 

Gold Top

1974 Yellow Submarine - inlay / tape
 

Compilations 1973-1982

1973 The Red Album - inlay
1973 The Blue Album - inlay
1973 Magical Mystery Tour - inlay
1976 Rock 'N' Roll Music - inlay / inlay reverse / tape
1977 The Beatles At The Hollwood Bowl - inlay / tape
1977 Love Songs - inlay / inlay reverse / tape
1979 The Beatles Rarities - inlay / inlay reverse / tape
1979 Hey Jude - inlay / tape
1980 The Beatles Ballads - inlay / tape
1982 20 Greatest Hits - inlay / inlay reverse / tape / alternate inlay (different EMI/Parlophone logo's)
1982 Reel Music - inlay / inlay reverse / tape

XDR versions

A Hard Day's Night - inlay (Apple re-issue) / inlay reverse (Apple re-issue)
Revolver - inlay / inlay (Apple re-issue) / tape (Apple re-issue)
Sgt. Pepper - inlay (Apple re-issue) / inlay reverse (Apple re-issue)
The White Album - inlay 1 / inlay 1 reverse / inlay 2 / inlay 2 reverse / tape

Budget label re-issues

1980 Rock 'N' Roll Music Volume 1 [MFP] - slip case / tape / wraparound re-issue / wraparound re-issue reverse  / inlay 2nd re-issue / inlay reverse 2nd re-issue
1980 Rock 'N' Roll Music Volume 2 [MFP] - slip case / tape / wraparound re-issue / inlay 2nd re-issue
1983 Oldies But Goldies [FAME] - inlay / tape
1984 At the Hollywood Bowl [MFP] - inlay / tape

Miscellaneous

1969 George Martin - Off The Beatles Track - inlay / inlay reverse / original green label tape
1972 Franck Poursel - Meets The Beatles (EMI Columbia release of orchestral instrumentals, psychedelic cover!) - inlay / tape
1975 "The Beatles on Tape" Catalogue - outside / inside
1975/76 "EMI Recorded Tape Catalogue" (packed into a case the size and shape of a cassette case, the pages 'fan out' from a hinge in the corner) - front opened to show Beatles pages / reverse
1979 The Beatles Concerto (Sleeve notes on reverse by George Martin) - inlay / reverse inlay / tape
1980 The Beatles Box -
inlay 1 / inlay 1 reverse / inlay 2 / inlay 2 reverse
inlay 3 / inlay 3 reverse / inlay 4 / inlay 4 reverse
inlay 5 / inlay 5 reverse / inlay 6 / inlay 6 reverse
inlay 7 / inlay 7 reverse / inlay 8 / inlay 8 reverse
1984 Their Greatest Hits - inlay / inlay reverse  / tape
1984 The History of Rock Volume 26 [Orbis Collection] - inlay / inlay reverse / tape
1986 Only The Beatles - inlay / inlay reverse / card / card open / card (version 2) / card (version 2) reverse
1987 All You Need is Love [Cassette Single] - inlay / tape

1990's/2000's Cassettes

1991 The Beatle Singles Collection on Cassette -
photo of cassettes in presentation box / 01-08 front / 01-08 back / 09-16 front / 09-16 back / 17-22 front / 17-22 back / tape example
1993 The Red Album [Re-issue] - inlay* / inlay reverse* / inlay 2*
*These are partial scans, the sections with the lyrics are not included
1994 Live At The BBC - cover / inlay 1 / inlay 1 reverse / inlay 2 / inlay 2 reverse
1995 Baby It's You [Cassette Single] - inlay / tape
1995 Free As A Bird [Cassette Single] - inlay / tape
1995/96 Anthology Albums -
Anthology 1 inlay 1 / inlay 1 reverse / inlay 2 / inlay 2 reverse
Anthology 2 inlay 1 / inlay 1 reverse / inlay 2 / inlay 2 reverse
Anthology 3 inlay 1 / inlay 1 reverse / inlay 2 / inlay 2 reverse / tape
Anthology 1, 2 & 3 in boxes with stickers
1996 Real Love [Cassette single] - inlay / tape
2000 "1" - inlay / inlay reverse

Thanks to Paul Humbley for the George Martin inlay scans, Rubber Soul, MMT, Their Greatest Hits, The Singles Box Set, Baby It's You and Real Love, to Peter Coombes for Abbey Road, to Gary for the MFP Hollywood Bowl, the 1993 Red Album re-issue, XDR AHDN, Revolver & Pepper, The Beatles Box scans, Live at the BBC & "1", to Mark Boulton for the original Please Please Me, Beatles For Sale, Help!, Rubber Soul, Revolver, Abbey Road and Let it be, the 1969 edition of Sgt. Pepper, the gold versions of Revolver & White Album, The Beatles Concerto, the Franck Poursel tape, the green label "Off The Beatles Track", "The Beatles on Tape" catalogue and the mid 70s EMI Tape Catalogue, to Ian Hossack for the original White album, gold Pepper, gold Let it be, Rock n Roll Music (& the MFP re-issues), Hollywood Bowl, Love Songs, Reel Music, the Orbis tape, the Fame Oldies but Goldies and the All You Need is Love single and to Henry Bell for the original pressing of "Sgt. Pepper", 20 Greatest Hits and Free As A Bird.