Tea is not very viscous (as in it is not very thick/sticky consistency) so when you turn the cup, the liquid stays in place, while the outer rim (the cup) rotates. If you did the same thing with a cup of honey, the hair would rotate. You can make the hair rotate in tea by doing repeated circular motion movements (imparting rotational momentum in the liquid over time) or by rotating the cup really slowly so that the liquid flow doesn't detach from the surface.