Can a beyblade make a weak tornado?

Not a real tornado. In order to form a tornado, even a weak one, there needs to be a complex setup of air currents withing a strong thunderstorm. By definition a tornado extends from the cloud base of a thunderstorm to the ground, a distance of hundreds if not thousands of feet.

A Beyblade is just a fancy top. Spinning it may produce some minor air perturbations, but these will not be noticeable at a distance of more than a few inches.