What happens to those who die and do not believe?
It depends. If you mean, “What happens to those who die rejecting Christ?” the Church’s answer is uncompromising: They will go to hell. But no one goes to hell by accident. If someone is simply ignorant of the name of Christ through no fault of his own, there is no sin in that. He has not rejected Christ. Moreover, we know Christ is not constrained by our knowledge. He can work in a heart even when that heart is only dimly aware of it.
Jesus speaks of this in the Parable of the Sheep and the Goats. The ones judged here are “the nations”—those outside the visible communion of the Church. How are they judged? By the way they responded to Christ when he came to them in disguise. “I was sick and you visited me; hungry, and you fed me; thirsty and you gave me something to drink.” How do the sheep and the goats respond? With surprise. The point is, we may not know it, but Christ comes to us even when we don’t (or can’t) come to him. This is why the Church counsels us to hope and pray for the dead. We must not pretend we know what God is up to in the lives of others. We know where the Church is. We do not know where it is not.