Effect of Hanning Window on phase of FFT
It't really more of a question of what the rectangular window is doing to your phase relationships, because it doesn't deemphasize the "edge effects" that occur at the beginning and end of your block of samples, which is the whole point of using a non-rectangular window.

Try this experiment: Do a number of FFTs with lengths ranging from 59*48000 samples to 60*48000 samples, using both window types. You'll find that the Hanning window gives much more consistent results than the rectangular window does. The two results will agree best when the length is close to an integer number of beats.