The amount of heat lost by evaporation and sweating never exceeds the amount of heat gained by the hot tea you drink. The extra heat makes the blood vessels near the skin dilate to help cool the blood faster. The nerves on the skin senses this and the body feels flushed and warm.
The answer is that hot tea will make the body sweat more,thus increasing cooling. However,the amount of extra cooling will not be enough to counteract the heating up from the tea.
So if someone tells you to drink hot tea in the summer, don't listen to them. Just pour yourself a glass of iced tea, iced water or cool lemonade.