The tiger comes out of the door, but it only does so after the story ends. Within the time-frame of the story, the door opens, and the audience is left waiting to see what will emerge. The author then describes the thoughts of the semi-barbaric princess, who has had to decide whether her lover will find happiness with another woman or be torn to pieces by a tiger.
The idea of the tiger killing the young man is admitted to be terrible, but the terror is dismissed in a brief paragraph consisting of a single sentence. A much longer paragraph then examines in detail how the princess had "gnashed her teeth, and torn her hair" at the thought of her beloved's relief at the joyful life he would lead with the lady, and how they would be married amidst general rejoicing:
Would it not be better for him to die at once, and go to wait for her in the blessed regions of semi-barbaric futurity?
It is clear which of the two options is the most horrifying for the princess. She, like everyone else in the story, is a flat character, and there is no complexity in the way she is drawn to suggest that compassion would win out over jealousy. Clearly, she chooses the tiger.
Or, perhaps not. The whole point of Stockton's story is that even the title is in the form of a question, intended to encourage debate. The case for the tiger is outlined above, but perhaps you can make a more compelling case for the lady. In either case, the reading and the discussion constitute the point.