I've had a thing for years about asking people if they mind if I sing their songs before I do them. At the same time, I suffer from acute embarassment if called upon to do it in their presence - even if it is someone I regard as a friend. On the other hand, it recalls an occasion 25 years ago when I went to see Tom Gilfellon in Nottingham, who had one solo album out at the time. All the floor singers sang some material from the album, and not always well - I was left wondering whether it was a tribute, or whether it was a conspiracy! Tom gave a marvellous performance, and did not touch the album material, so it did not bother him too much, but it would not have been kind if he was trying to plug his record, or a person of restricted repertoire.
I know one girl who sounds so much like Joan Baez that you don't need the original. For myself, when I start fading halfway through a gig I get people coming up to me saying I sound like Howling Wolf (which would be OK if I wasn't trying to sound like Big Joe Turner). I don't think it is harmful if you sound like someone famous - but in general performance terms, you either have to sound exactly like the original, or be completely different, to be acceptable to the listening public.