I mean, he could still lie. He’d just have to afford one more question

Because there are two doors and only one question. If you ask a known question unrelated to the door you find out who the liar is but lose your opportunity to ask them which is the correct door.

That assumes the other guy holds to his principles in the face of death. If I were the dm, the act of tearing the other guy’s head off and then threatening to do the same to the other one unless granted another question would at least grant advantage on an intimidation check