The first sentence, "I don't need you answer my question." is incorrect. In order for it to be correct, you would need to add the word "to" before the word answer to make it the infinitive form of the verb "to answer." This is exactly what you did in the third sentence, and it is correct. The second question is correct as well, but this implies that you generally don't need them answering your questions, not just on this one occasion.
For example, if you asked your friend a question, but someone else answered instead, you could say, "I don't need you answering my question." You're implying that you don't need them to answer your questions.
However, if you were to respond with, "I don't need you to answer my question." It's generally taken to be just this one occasion. You don't need them to answer this one question.