The cover of “Gimme a Stone” that is on the album Largo (various artists) is better by far than the original by Little Feat.
Stone Temple Pilots cover of Led Zeps Dancing Days.
They changed the tempo a little bit I think… made the song perfect.
Stairway to Heaven covered by Rodrigo Y Gabriella
Here, There and Everywhere covered by Emmy Lou Harris
Yeah, love their acoustic version. Talent in that band.
Mmm, quite a few Led Zeppelin songs. 
Eva Cassidy’s version of “Time After Time.”
Steve Lawrence and Eydie Gorme’s version of Black Hole Sun.
the beatnix cover of stairway to heaven
Satisfaction by Devo is one of my favorites. Nothing like the original, but oddly funky.
Got To Get You Into My Life by Earth Wind & Fire. Tough to find covers of Beatles songs that are better than the original, but I love this one. Also a contender is Aerosmith’s version of Come Together, but maybe the original edges it out. Aerosmith certainly pack the sleaze onto it, though.
Anything by Manfred Mann that Springsteen or Dylan wrote.
And a second vote for Hurt.
Amen.
Also: Bobby Womack’s “It’s All Over Now” was redefined by Sir Michael and band. In fact, it remains one of my two favorite Stones tracks.
Harry Nilsson’s cover of “Without You” (by Pete Ham and Tom Evans of Badfinger).
The way Dylan and The Band play it on *Before The Flood *seems to support this.
A lot of people will say the Johnny Cash version of “Ring of Fire,” but great as it is I give the edge to the original for Anita Carter’s sublime voice.
Mr. Tambourine Man by William Shatner? 
On a serious note , many songs by Eva Cassidy. People Get Ready is one of her best.
Turning back the clock a bit…
Turn, Turn, Turn (To Everything There is a Season): I like The Byrds’ version better than all earlier ones.
Lily, Rosemary and the Jack of Hearts: I think Joan Baez brings more of a depth to the song that Dylan did.
Sting’s “Little Wing” totally kills Jimi’s.
I like Bett Midler’s “In My Life” better than the Beatles original.
People Are Strange by Brave Combo