The Beatles, surely the most famous band with a pun-based name.
Also Rubber Soul, a pun on rubber soles in shoes and the phrase "plastic soul", which Paul McCartney found out was how some black entertainers called the white blues-rock of The Rolling Stones.
Led Zeppelin (kind of a pun on "lead" as in "you'll go down like a lead balloon", said by Keith Moon to Jimmy Page meaning It Will Never Catch On); actually it was changed to make the pronunciation unambiguous.
Their reggae parody "D'Yer Mak'er" is a pun on "Jamaica", but ultimately it's based on an old joke:
My wife's gone to the West Indies.
No, she went of her own accord.
And speaking of reggae, let's not forget the parody band Dread Zeppelin.
The rock group Slade has an album called Slayed?. This was a reference to their frequent use of misspelt titles, eg "Take Me Bak 'Ome" and "Gudbuy T'Jane".
The Rush album Moving Pictures might be considered one twice over, since on the surface one would think it refers to movies, but the cover shows people carrying paintings out of a museum, as well as people crying or appearing otherwise emotionally affected—that is, moving the pictures out of the building, and also moved by the pictures.
The VAGIANT (now Tijuana Sweetheart) album Public Display of Infection.
Their old name also qualifies, being a junction of "vagina" (it's an all-girl band) and "giant".
The Greg Kihn Band apparently really like making puns on Greg Kihn's last name. Their discography includes albums called Next of Kihn, Rockihnroll, Kihntinued, Kihnspiracy, Kihntagious and Citizen Kihn. Only three of their studio albums don't involve a freakihn' pun of some kind.
"Animal Nitrate" by Suede is a play on the drug amyl nitrate (and the animalistic nature of the protagonist's Bastard Boyfriend).
This Heat's album Made Available: It's a collection of sessions they recorded for the BBC at Maida Vale Studios. They also punned on their own band name by calling another album Deceit.
The one-hit wonder band Lipps Inc. of "Funky Town" has a slightly self deprecating punny name if said aloud.
The Cranberries were originally The Cranberry Saw Us (you know, like "the cranberry sauce"). Once vocalist Dolores O’Riordan joined, she convinced the others it was a particularly groanworthy pun and they shortened it to just The Cranberries.
Jean Michel Jarre's Les Chants Magnétiques: literal translation = "The Magnetic Songs", a pun on "champs magnétiques" = "magnetic fields". The album was released as Magnetic Fields in anglophone countries.
Kristin Hersh punned on her own name with the song title "Christian Hearse".
John Wilkes Booze, a pun on Abraham Lincoln's assassin John Wilkes Booth. Better yet, they were originally called The John Wilkes Booze Explosion, a pun on both John Wilkes Booth and the name of another band, The Jon Spencer Blues Explosion.
Sting's Ten Summoner's Tales initially just seems like a reference to The Canterbury Tales, but it's also a pun on his real name, Gordon Sumner.
The Miller's Tale, a Tom Verlaine anthology, is based on the same pun as the Sting example.
Gratuitous Sax and Senseless Violins by Sparks.
The Mind is a Terrible Thing to Taste by Ministry. They're also fond of punning on classic rock album titles - see Rio Grande Blood, Houses of the Molé and Dark Side Of The Spoon. And they've even punned on their own album titles with a couple of remix albums: Rio Grande Dub Ya (referencing both the genre of dub and George W. Bush's nickname) and The Last Dubber (referencing their album The Last Sucker - already a pun on "the last supper" - and dub again)
Al Jourgensen is a fan of puns in general. Some of the tours he's done include the SphincTour, CliTourIs, FornicaTour, MasturBaTour, C-U-LaTour, LubricaTour, DefibrillaTour...
Satirical death metal band Faxed Head have a song called "Gore And Guts". Given the title and the genre, you'd expect a song involving gorn or a Shout-Out to the band Gorguts - it's actually about then-vice-president Al Gore ("Albert Gore has the guts / to lead this nation of god damn nuts!")
Alex Chilton's album Bach's Bottom puns on his old band The Box Tops.
Four Sail by Love. When you combine the album title with the band's name you get "love four sail", or "love for sale".
Road To Rouen by Supergrass, a pun on the expression "on the road to ruin" and Rouen, a city in France. It may also be a Shout-Out to the Ramones album Road To Ruin.
The Ned's Atomic Dustbin album God Fodder ("godfather"). Supposedly it was inspired by a pun made while the band were playing a round of Trivial Pursuit - one of the questions asked what "the food of the gods" was, and a member of the band jokingly gave the album title as their answer.
Parasites' debut album Pair Of Sides, playing off the band's name and the two "sides" of a record or cassette (the album was not initially released on CD, a format the pun wouldn't have worked for).
The Australian band X (not to be confused with the otherX or X Japan) released an album called Aspirations... As in "Exasperations".
The Vampire Weekend song "Diane Young". (As in "Dying young".)
Hepa-Titus (whose name is already a pun on "hepatitis" and possibly the Roman Emperor Titus) have a single with the title Don't Half A Cow. Aside from the expression "don't have a cow", the title references the fact that the A-side features half of the original lineup of the band Cows - the band's bassist is former Cows member Kevin Rutmanis, while former Cows vocalist Shannon Selberg makes a guest appearance.
From Garth Brooks: Chris Gaines' fictional Fornucopia album, which is a portmanteau of "fornication" and "cornucopia".
So far, Weedeater have done this with three out of their four album titles: ...And Justice for Y'all is a parody of Metallica's ...And Justice for All. God Luck and Good Speed is sort of a Spoonerism for "Good Luck and Godspeed", and also a likely reference to "speed" as in amphetamine. Jason... The Dragon is another drug pun, referring to "chasing the dragon".
The title of Yes' 1978 Tormato album was a reference to Yes Tor, a Welsh geological formation where Hipgnosis shot the original album cover, and the tomato thrown by keyboardist Rick Wakeman at the picture when disappointed with the cover photo; the combined results made the final print◊.
The Happy Mondays album Uncle Dysfunktional. From that album, there's "Anti Warhole On The Dancefloor".