Write an ending to the story. It may not involve opening doors.
What you need to examine are the possibilities of other things happening that have been foreshadowed in the story. You know that there is a tiger running around somewhere. Perhaps he/she got out? What would be an appropriate ending if this happened?
Another option is the fact that the princess really loves the young man. Readers know that she does everything she can to find out what is behind which door. Perhaps her father found out that she knew and tells her as the young lover is going for one of the doors. What would happen then?
Another option is the princess could have told her father that she really loved him. The princess could have told her father that she would do anything for him. What do you think the king would have said? What could the king have done?
What about the young lover? Have there been any hints as to what happens if the accused refuses to play the king's game? What if he does?
I really hope one of these prompts help you to write your own ending.