I personally prefer my characters mortal, and for that reason I missed much of the joy of reading superhero comics as a kid; as a small boy, I simply didn't want to read about heroes who either couldn't die or couldn't stay dead (that's a little ironic, considering my religion, but what can I say? I'm still of the opinion that, in fiction, most resurrections are cop-outs: I just don't believe that Nausicaa can be resurrected by giant millipedes or that Neo can be resurrected by nothing in particular).
Over at SF Signal, they are discussing a slightly different problem: how to create drama in a far-future universe where the problems of death or scarcity have been licked. Now, I personally don't believe these are problems that will ever get licked, and I'm fond of the infinite razorblade parody of the technological singularity, but supercivilizations full of decadent immortals are so common in far-future sf, the problem has to be dealt with. Several authors have weighed in on the question:
Post-Scarcity and Post-Singularity, Part 1
Post-Scarcity and Post-Singularity, Part 2
John C. Wright, who it's obvious by now I admire, has a particularly neat answer to the question, which he makes in reference to his Golden AgeTrilogy: a post-scarcity society is simply impossible.