Edit: Thanks for the Allegaeon and Prong ones. Those rock!
Chris Cornell - One (U2 with Metallica's lyrics)
Big Head Todd and the Monsters - Tangerine. BHTM has a lot of great covers, but Tangerine's where it's at.
Living Colour - Memories Can't Wait
Jack White - Mother Nature's Son/That Would Be Something - Look up the video of this one. Performed it with McCartney and Obama literally right in front of him. Balls.
Sheryl Crow - The First Cut is the Deepest
Local H - Hackensack, When Doves Cry (both versions) Again so many good ones
The Bangles - Hazy Shade of Winter
Anthrax has so many. Some of my faves are - Lookin' Down the Barrel of a Gun, No Time This Time, Sabbath Bloody Sabbath and Neon Knights. Their Kiss and Cheap Trick ones are awesome but basically just amped up versions of the originals.
Gipsy Kings - Hotel California
Les Claypool's Duo de Twang - Stayin' Alive
Dynamite Hack - Boyz-n-the-hood
Bad Wolves - Zombie
Eddie Vedder - You've Got to Hide Your Love Away
Nirvana - Turnaround
Soundgarden - Girl U Want
Michael Andrews and Gary Jules - Mad World
Pearl Jam - Love Reign O'er Me
Soul Asylum - Sexual Healing
Willie Nelson - Blue Hawaii
Too many