Another good one, is from the tribute album to Warren Zevon, called "Enjoy Every Sandwich". Jackson Browne, and Bonnie Raite do a cover of "Poor Poor Pitiful Me".
Yeah, great song. Always forget it's a cover though. Only reason it occurred to me was that I was listening to my digital copy and the infobar will occasionally display "(tears for fears cover)". That made me think of this thread so I youtube'd it then tossed it up.
Two that come to mind are Johnny Cash's rendition of Hurt (obviously) and Sarah McLachlan's version of Dear God, by XTC. Both covers I like better than the originals.