What happens to Christians who are still alive when the rapture occurs?

Christians alive at the rapture will not die and be resurrected; instead they will be changed in a moment, in the twinkling of an eye, at the last trumpet. Their corruptible body puts on incorruption, and their mortal body puts on immortality, becoming flesh and bone activated by the Spirit.

Les explains that this event has no prophetic sign, so no one can know the day or year. At the last trumpet, the dead are raised incorruptible, and those who are alive and remain will not die and be resurrected; instead, they will be changed. In a split second, their body of corruption becomes a body of incorruption. This corruptible must put on incorruption, and this mortal body, prone to death, must put on immortality. It loses the flesh and blood concept and becomes flesh and bone activated by the Spirit.

1 Corinthians 15:51

Behold, I shew you a mystery; We shall not all sleep, but we shall all be changed,

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Will Christians who are alive when the trumpet sounds have to die first?

No, believers alive at the trumpet will not die first; they will be changed in a moment, in the twinkling of an eye, at the last trump. The dead are raised incorruptible, while the living are changed from corruption to incorruption directly.