When you wash your hands, which water do you use: hot or cold? Most people use the hot one. How often do you actually run the hot water long enough to get hot water?
This was a habit that we didn't notice until we got a tankless water heater. The concept is great: With a larger burner, generate hot water on demand instead of keeping ~30 gallons of water hot all the time. Out for the week? Doesn't matter, zero gas used. Another advantage from a usability stand point is that we can take two hot showers at the same time. Forever. (Note: The temp does fluctuate, so getting a thermostat controlled shower fixture was highly recommended. I love them too)
So every time someone runs some hot water, there is a pressure drop at the heater, and it lights up in all of it's 50,000 BTU glory and heats the water on demand. But if it's just for washing hands, and the water is never really run long enough, all this does is heat up some copper pipe in the wall: wasted energy.
There is nothing wrong with washing your hands with hot water. I enjoy the nice hot hand washing now and again, but if you're going to use the energy to heat the water (tankless or not), just try and use it.