I discuss this with friends and family occasionally, and they all seem to have these ideas that we're "only supposed to live 100 years or so". But that seems like a logical fallacy (appeal to nature) to me, it's not an actual argument.
And sure, there are plenty of dystopian movies about how all of your friends die while you live on. But you know, you can make new friends. As someone who has moved often and had to make new friends often: it's not that bad. Friendships come and go.
Just imagine the things you could get done if you weren't pressured by time. You could witness the colonization of other planets, the unification of humanity. And if everyone were immortal, we wouldn't have to worry about things like pensions anymore, and I bet that we'd see an extreme lowering of crime and violence.
The only read problem I see would be overpopulation, and possibly draconion measures to regulate it.