Johnny Cash also does a nice cover of Tom Petty's "I Won't Back Down". Little slower and plaintive.
Dylan's voice is so one of a kind that covering Dylan in a pure sense can almost be impossible. There are some decent covers of Dylan songs out there. The Byrds Mr. Tamborine Man, Chimes of Freedom, and My Back Pages. Hendrix's version of All Along the Watchtower is more well known than Dylan's own version. And The Grateful Dead and Jerry Garcia Band did countless covers of Dylan songs in their concert rotation. Usually Garcia or Weir would sing, depending on the style of song, except Lesh did sing Just Like Tom Thumb Blues.
One final song that is interesting is "Because the Night", which was co-written by Patti Smith and Springsteen. Many people consider Smith's version to be the original and Bruce's the cover, but each can claim credit. I actually prefer Springsteen's. One sings it from a man's point of view, and the other a woman's. Of course, Natalie Merchant's version is a cover.