Cher - All I Really Want To Do (Shindig! 1965)
Honestly I think some of my favourite Bob Dylan covers (besides those by Baez, of course!) are the ones that Cher did on her first two albums. According to her memoir Dylan said he quite liked them as well, so that's pretty cool. Sonny and Cher were pretty hip and cutting-edge in the 1960s, I think a lot of people tend to forget that. But they were some of the first hippies.
The Byrds covered several Bob Dylan songs including Mr. Tamborine Man, All I Really Want to Do and You Ain't Goin' Nowhere. Here's what Roger McGuinn said about Bob Dylan's feelings on the matter in an interview with Rick Landers (http://guitarinternational.com/2010/10/21/roger-mcguinn-talks-the-byrds-martin-guitars-and-home-recording/
):Rick: Did Bob Dylan ever say anything about your renditions of his music?
Roger McGuinn: Yeah, in fact he was at our rehearsal hall when we were working up the song “Mr. Tambourine Man.” Not only that song, but we did quite a few of his other songs.
I remember playing, I think it was “All I Really Want to Do” and he asked, “What was that?” I told him it was one of his songs. And he said, “Ah ha!” He didn’t recognize it. Bobby Neuwirth was there and said, “You can dance to it, that’s pretty cool!”
And they were very supportive of what we were doing with Bob’s music. So, we recorded “Mr. Tamborine Man,” and Albert Grossman tried to stop it. Bob was behind it, but his manager wasn’t. But, it had already gained enough momentum that Columbia couldn’t stop it.
Bob Dylan wrote You Ain't Goin' Nowhere in 1967. His original lyric had the line "Pick up your money and pack up your tent." When The Byrds covered the song, their lyric was "Pack up your money and pick up your tent."
In 1971, Bob Dylan released You Ain't Goin' Nowhere on his Bob Dylan's Greatest Hits Vol. II album. In mock annoyance, Bob Dylan sang "Pack up your money, pull up your tent McGuinn / You ain't goin' nowhere."
In 1989, Roger McGuinn replied "Pack up your money, pick up your tent Dylan" on a new recording of the song included on the Nitty Gritty Dirt Band's Will the Circle Be Unbroken: Volume Two album.