Weird Al. His polka medleys always make me smile. 
“Baby Got Back” Jonathan Coulton
“Can’t Get You Out of My Head” The Flaming Lips
Not sure if “ironic” describes this one, but “weird” might fit the bill…
Manfred Mann’s Earth Band’s cover of Bruce Springsteen’s Blinded by the Light.
Dramatic change in instrumentation, beat, tone, and atmosphere.
Oh, and one of my favorites…
Tom Jones and the Art of Noise’s cover of Prince’s Kiss.
Awesome.
I never thought much of Tom Robinson, never thought he had any real talent, and didn’t like his rendition of Steely Dan’s “Ricki Don’t Lose That Number.”
But hearing that song performed by a gay man gives the lyrics a whole new meaning.
And one more, another favorite of mine.
Gary Jules’ cover of Tears for Fears’ Mad World, which played over the final credits of the movie Donnie Darko.
Jules’ version is beautiful, and while i liked the Tears for Fears original, the cover matched the music and mood much more closely to the lyrics than the original did, IMO.
The Residents - “Hitler Was a Vegetarian” is a medley of several songs. They also covered The Stones’ “Satisfaction,” “Paint It, Black” and Prince’s “1999.”
Good lord, who hasn’t covered that? I have versions by Bowling for Soup and Nicotine.
Anything by Dread Zeppelin. I have fond memories of them on a double bill with Mojo Nixon: Jesus, that show was fun.
I think Travis covered it too.
Clem Snide doing Christina AGulera’s Beautiful
Aztec Camera doing Van Halen’s Jump
I was always tickled by The Vandals’ very faithful cover of Queen’s Don’t Stop Me Now
The entire Me First and the Gimme Gimme’s catolog
Luther Wright and the Wrong’s Bluegrass album Rebuild the Wall (a front to back coiver of Pink Floyd’s the wall)
Hayseed Dixie and their bluegrass covers of Kiss, AC/DC and other things
Richard Thompson has done any number of very odd covers, including: Oops, I did it Again, Ce Plane Pour Moi, *Heard it Through the Grapevine * (with Fairport Convention and Roy Wood’s Big Band) and, (as part of French, Frith, Kaiser, Thompson) Surfin’ USA.
Some of my favorite covers:
- The Clash’s “Straight to Hell” done by Moby and Heather Nova
- Pink Floyd’s “Comfortably Numb” by the Scissor Sisters
- Billie Holiday’s (sorta, it was also a cover) “Gloomy Sunday” by The Smithereens
- The Rolling Stone’s “Paint It Black” by Gob
- Schoolhouse Rock’s “The Tale of Mr. Morton” by Skee-Lo
And speaking of Britney Spears covers, Marty Casey did a fabulous version of “Baby One More Time” on Rock Star: INXS.
I like the Blues Brothers rendition of that song, too, but their version of “Stand by Your Man” is truly one for the ages 
Also, although not a cover, I just thought of an unintentionally ironic usage of a song. During the opening ceremonies to the Torino winter olympics, various American pop/dance songs, mostly from the '70s and '80s, were played as the national teams promendaded in. This in itself was weird enough. However, the American TV analysts at least kept injecting various factoids/commentary about each nation as their team marched in, many of them sad. This reached a crescendo when the Bosnian-Herzegovenan team marched in and the analysts started discussing the tragedy that that country has gone through, and specifically noted with sadness the 1984 winter games which were held in Sarajevo, that city being a scene of horrific violence and cruelties during the war of the early '90s.
The song playing in the stadium at the time? “Funkytown” by Lipps, Inc. I swear, you can’t make this stuff up.
Is that the one on the extras disc? I loved that! I actually sing that version rather than the original!
I’m not sure - which extras disc? I know it from the “Schoolhouse Rock Rocks!” CD, which has covers by Moby, the Indigo Girls, etc.
And I actually prefer this version as well.
Abso-smurfly! Their version of Spinal Tap’s “Big Bottom” is amazing.
I took young Miss Greenjeans (age 8 at the time) to see Hilary Duff a few years back. The encore was a bubblegum version of “My Generation.” I chuckled to myself when the singing teen hoped she died before she got old.
I rather like Devo’s cover of Ohio. Setting it to a martial beat seems ironic.
I’m not sure what Scissor Sisters meant by their Comfortably Numb, but I got a chuckle out of it.