You should ask a tougher question.
The play was perfect, you put him accurately on a range of hands that made it prudent for you to check behind him on the flop, and you got all in when you had the best of it, with a push that would put your opponent in the chip and a chair position if he lost. You would rather have a call here from your opponent, but only if he calls all you got to bet. If he folds it is just fine, your tournament precedes after that hand. But doubling up at this point, is worth the risk you took.
I would also ask what the average chip stack was at the table. Four handed and you and this opponent had the vast majority of chips, you should not of played the hand at all. If the tournament only played three spots you should not of become involved in the hand.
Let me clarify what I am getting at in the italic. He made a good play pre-flop. He raised the hand, which is what you simply do with almost every hand you play at a four handed final table. Even with marginal hands. The thing that happened is that he was re-raised.
At this point there are a lot more things to consider. The play has gone wrong. He has not succeeded in taking the blinds, which would be the A plan, and part two taking a little more with a follow up bet, is no longer likely to happen.
Generally speaking he should call, the villain made a huge mistake by making his raise to small. If your going to reopen the action with pocket kings, you should make an assumption that you dominate and raise enough to pick up the dead money, something like three to five times the original bettors raise, or you should not raise at all and trap your player on the flop. In a tournament I would prefer to make a big reraise with Kings, in a cash game out of position I would lean a little more toward calling depending on the raiser and the field.
Hero generally has a calling hand here, he is getting a little over three to one on his call, the call is not crippling to his stack and he has position. However there are reasons to fold at this point also. The amount he has invested is small and his hand against a raise from a tight player is at best perilous to his tournament life. His hand of Queen-Ten is no better then 5-7 against the rank one hand he suspected the player had. He needs a strong flop, two pair, three of a kind or straight, or a straight draw and hope he does not have to waste to many chips missing it. Then hero gets to bust a player and go merrily on his way to win the tournament.
But those kind of flops do not happen very often. What happens more often is what did happen, one pair, top pair on the flop is hit, the queen in our hero's case or the flop is missed entirely. If a hand that hits is still unplayable after the flop, I have base argument to muck the hand before the flop. Or in other words I need compelling reasons not to muck the hand before the flop.
In the case in question there might of been enough reasons, mainly the low price compared to the stack, a little overlay and position. However, if any of the following conditions had been true I would of let Q-10 go when re-raised.
If the Villain had made more then a doubling raise I fold.
If my stack was much smaller, I fold.
The longer the tournament rounds, the more I would tend to fold here.
If the villain was anything more then tight weak I would fold.
Being in position at a low price with a hand that needs help is not a bad thing. But if it is going to cost you, and you are going to be able to find better spots, it is often better to let this hand go. The real negative about playing this particular hand is that the villain is going to be able to bust the hero out much more often then he is going to go broke in the situation.