What kind of immortality are we talking about here? Can we still get sick? Can we grow old? I think if it was immunity to sickness and age, it would mean life would get boring pretty fast. A lot of things to think about like can the human brain remember things from 1000 years ago?
A person will turn into a robot if he does not hurt and memorize a large amount of information.
I think that the mechanism of death is to not overpopulate the earth. This is very important. Even if humans were immortal they would have fought for every piece of land and would still be killing each other.