They don't kill the potential Ones, but they all typically end up dying anyway.
What Cypher is talking about is Morpheus plan to train Neo and send him against the agents; because the Agents are programs, they can break the rules of the system better than any plugged-in human, no matter how well trained. Only The One, with his implicit control over the Matrix itself, can possibly stand up to them.
They didn't talk about it much in the movie, but from the supplemental materials (and previous versions of the script) we know that Morpheus has found a number of former One candidates, and done basically the same thing. In fact, Cypher is one of these former potentials, one of the few still alive, which is partly why he's so cynical: he's heard Morpheus pitch before and it turned out to be BS, at least for him.
If Neo had not been the One, and he had tried to rescue Morpheus from the Agents, there's almost no chance he would have survived. He almost didn't survive, since the Agents did kill him before he recovered. Cypher was convinced that Neo was just another false candidate, and that Morpheus was going to send him into the Matrix where he'd get himself killed.