Do you need to leave hot and cold water on to keep it from freezing?

Given that you are not changing the fluid in your water heater, and your heating source is 100% efficient, and you are paying a constant rate for your heating source, then turn off your water heater. Reason: The water in your water heater will always try to return to room temperature. The rate it returns to the room temperature is governed by the insulation around the water heater and the difference in temperature between your set point and the room temperature. So to keep the water heated to the set temperature, it requires adding the heat that passes through the insulation over time. If your water cools some amount, lets say 10 degrees, there exists a smaller temperature difference between the water temperature and the room temperature, and the cooling rate slows. This is why there is savings if you turn down your room temperature in the winter, or the water temperature.