From a Physicist's point of view, the amount of heat that must be replaced, if you ignore the hot water that is removed from the tank, is dependent on the difference in temperature of the water and the temperature of the surrounding air.
If the heater is left on, the heat loss over time stays pretty well constant. If the heater is turned off when not in use, the water cools and the heat loss is less.
So, the amount of propane used to replace the heat loss is less when the water is allowed to cool for a period of time, even counting the propane necessary to reheat the cooler water.