I love Rush's version of the Yardbirds' "Heart Full of Soul" (on Feedback), which Geddy apparently thinks is one of their best recordings. It towers miles above the original.
But my favorite is probably Jason & The Scorchers' blistering version of John Denver's "Take Me Home (Country Roads)" (on A Blazing Grace). It just rips. A few years ago, my band shared a bill with a band that played Jason's version of "Country Roads," and they were shocked I recognized it. They clearly hoped folks would think it was their own adaptation, but Warner Hodges's guitar intro gives it away.
Also Roxy Music's version of Dylan's "A Hard Rain's Gonna Fall." I love Bob Dylan's version, but Bryan & Co. did wonders with it.
Though I won't say it's better than the original, I love Phish's live version of the theme from 2001: A Space Odyssey.