Nothing. Being in the dead pile just means you can't Marshal a copy of that character into play, it has no effect on an extant character or attaching a dupe to that character.
Hmmm, I thought I read somewhere that you cannot marshal a dupe if that character is in the dead pile(which is different from first edition). I'll have to check the RRG again when I have some time later, but you're probably right. It may have just been someone else's incorrect comment that is just stuck in my brain.