I don't really see this as being a problem at all, you just end up telling stories where fear of death isn't a motivating factor in how the stories turn out. I think its place would be taken by other dire consequences (losing your soul, perhaps? horrible mutilation? I dunno), and that the focus of the stories created would probably be more on what difficult achievements you get, and perhaps most importantly, on your achievements *relative to others*.
Old age is still an issue, but that's probably outside the scale of most games.
Hmm, I just had a thought I didn't start this reply with. If violence does not deliver the finality of results it normally does, it kind of puts it more on par with other problem solving approaches. For one thing, you can't finish an opponent by killing him, so you might as well persuade him or evade him or whatever. On the other hand, since he'll come back from it, it's also less of a big deal, so stabbing that guy to show that you didn't appreciate his review of your play might not be regarded as an overreaction.
I think all of this would make things interesting, if perhaps a little more 'light hearted' (in a dark way) than regular AW where death is a very real and constant threat.