I hate to give a starving dog a rubber bone, but here are two stories behind original versions that I can't track down audio of:
Because The Night covered by Patti Smith
The original song was recorded by Bruce Springsteen during sessions for his Darkness on the Edge of Town album. He was not satisfied with it; the Patti Smith Group was working on Easter in the studio next door, with engineer/producer Jimmy Iovine working on both albums. Iovine gave Smith a tape of the song, she recast it, and it was included on Easter, becoming the first single release from that album. Though it was never released on any of Springsteen's studio albums, in concert beginning with his Darkness Tour Springsteen would often perform the song with his own lyrics. The song was first performed live (with Patti Smith) at CBGB's in New York City on December 30, 1977. Until the 2010 release of Springsteen's box set The Promise, which included the original recording from the Darkness on the Edge of Town sessions, the only commercially released recording of a Springsteen version of the song was in the 1986 box set Live/1975–85, where Smith was listed as co-writer.
Private Dance covered by Tina Turner
The song was originally going to be included on the Dire Straits album Love Over Gold. After the song was recorded, however, the vocals were not added. Mark Knopfler considered that they were not suitable for a male to sing, so the track was cut from the final mix. Legal restrictions prevented the original recording from being used, so two years after it was recorded, the track was remade by Dire Straits for Tina Turner. The original drummer Pick Withers had been replaced by Terry Williams. Also, Knopfler did not feature on the track and was replaced by Jeff Beck.