I can't think of many examples, but it must happen. I can think of the Kaiser Chiefs covering The Slits version of 'Heard it Through the Grapevine' intead of Marvin Gaye.
Primal Scream used to perform the Jagz Kooner remix of Swastika Eyes live, and !!! used to perform a remix of Intensify when they played.
any other examples? Or am I banging my head against a brick wall?