Old cover by Nilsson…
Newer cover by The Lumineers... The pumps don't work 'cause the vandals took the handles.Fairport Convention’s “If You Gotta Go, Go Now (or Else You Got to Stay All Night).” It’s in French.
Nanci Griffith, Boots Of Spanish Leather
Tracy Chapman, The Times They Are A Changing
Johnny Winter, Highway 61
One More Cup Of Coffee The White Stripes
Don’t Think Twice It’s Alright Mike Ness
My 3 suggestions also played their versions for that show! (My Chapman link is from that show.)
Lots of great suggestions so far, but I’ll give a general suggestion for the soundtrack of “I’m Not There”, it’s great throughout and has a very special, but satisfying choice of songs. There are a lot of standout tracks, e. g. I’m Not There by Sonic Youth, Goin’ To Acapulco by Calexico or Cat Power’s Stuck Inside Of Mobile With The Memphis Blues Again. But they’re all great.
You saved me the trouble.
I did use the Byrds, but went with: “Wheels on Fire”.
Peter, Paul, and Mary: "Don't Think Twice" was the one I went with.“If you Gotta Go”-used Manfred Mann.
“One Too Many Mornings”-Used The Association version.
And If I included “Chimes of Freedom” these days I’d go wit the Boss.
Shown here in East Berlin.
Bryan Ferry’s A Hard Rain’s Gonna Fall is a bit of an acquired taste, but I happen to like it. It’s certainly very Bryan Ferry.
He also did a pure cover album called “Dylanesque” which is not bad at all.
“Leopard Skin Pillbox Hat”. Old Crow Medicine Show
Check out their album “50 Years of Blonde on Blonde.”
Need to change my post count.
Here’s a couple of really old covers…
“Like a Rolling Stone” – Jimi Hendrix
“It Ain’t Me Babe” – the Turtles
“I Shall Be Released” – the Band and damn near everybody from The Last Waltz
“Mr. Tambourine Man” – the Byrds
“I’ll Be Your Baby Tonight” – Linda Ronstadt
“Just Like a Woman” – The Concert from Bangladesh. I know Dylan plays on this, but he is backed by Leon Russell, George Harrison, and Ringo Starr.
“Knockin’ on Heaven’s Door” – Eric Clapton
“If Not for You” – George Harrison
“Desolation Row” – the Grateful Dead
“It’s All Over Now, Baby Blue” – Van Morrison
“Sad-Eyed Lady of the Lowlands” – Joan Baez
“I Shall Be Released” – Ladysmith Black Mambazo
"All Along the Watchtower – Jimi again
Naah, Sisters of Mercy
More from the more alternative side:
Wanted Man by Nick Cave & The Bad Seeds
Highway '61 Revisited by PJ Harvey
Wheel’s On Fire by Siouxsie and The Banshees
The Lonesome Death Of Hattie Carroll by Steve Howe.
Don’t know if it’s any good, but neither the song or artist has been mentioned yet, and I don’t want to piss off the OP. 
I did use Ronstadt’s version of “Baby”.
And don’t worry about pissing me off. It’s all for fun here.
“Open the Door, Homer” by Thunderclap Newman.
Since famously cited Odetta as the spark behind his desire to become a folk singer, you might maybe want to include one of her Dylan covers.