When you blow on a hot drink, you blow away the heat that has risen to the surface to help cool the drink down quicker. When you blow on your hands in winter,your exhaled air is warmer than outside air and so heats up your hands temporarily.
When you blow on a liquid such as hot tea you are actually helping the liquid to vapourise by blowing away molecules, preventing them returning to the body of the liqued. Vapourisation takes a great deal of energy (heat) so the drink cools.