I foolishly stuck my hand in the tank with a paper towel, not washing my hands first. Do I have to dump all the water out?

Im guessing that you are talking about a fish tank? How did you manage to stick your hand AND a paper towel in the tank by accident? Either way, no you don't have to change the water. Unless there was something bad on your hands. Like bleach or some other cleaning chemical.

Did your hands touch soap? Any potentionally dangerous chemicals?

If not then its fine. Ive done it many times.

If you did then I suggest a water change.

What was on your hand that you are so worried about?

If it was covered with used motor oil, dish washing liquid or peanut butter, then you have a problem.

If it was just a normal unwashed hand, no worries at all.

An aquarium is not a sterile environment, it's naturally full of bacteria and algae etc. A bit of germ transfer between you and the tank is nothing to worry about. Only time you should worry is if you have chemicals on your hands, makeup and hand cream could be a problem if you don't wash it off.

Ian