You only need to take out 1 marble.
The key to solving this riddle is knowing that every box is mislabeled. The first step is to take a marble from the box labeled BW. Let's say you draw a black marble, you then know that the other marble in the box must be black too, because otherwise the label would be correct. Now that you have identified the box with 2 black marbles, you can automatically determine the contents of the box labeled WW. You know it does not contain 2 white marbles, because the label must be wrong. You know it does not contain 2 black marbles, because you've already identified that box. Therefore, it must contain 1 black marble and 1 white marble. Then, of course, the last box must contain 2 white marbles.