"Do you want to listen to a story" - This one is correct. "Do you want to listen TO a story or not" - This one was almost correct. The difference between this one and the first one is that you would add "if not" if someone is unsure about listening to the story. For example, you have a child who tells you that he wants to listen to a story. After you start reading the story, he changes his mind and says he doesn't want to listen. Ten minutes later, he says he wants to listen to a story again. Then you could say "Do you want to listen to a story or not?!"

The first sentence is not correct but the second and third are correct.