Interesting question. In the games it is implied that Pokemon die at some point. There are places such as the Pokemon Tower in Lavender Town that have graves for Pokemon, and many trainers mourn the loss of their Pokemon.
There is also a myth in the original games that Gary's Raticate died in the battle before you see him at the tower, and he is visiting Raticate's grave. Though there is no real evidence for this.
However, in the games there is no way for your Pokemon to ever die. If they run out of HP they only faint, and will stay fainted permanently if you don't heal them. The only other way for Pokemon to disappear is to release them back into the wild, but of course they are still alive.