Hi,
I did a test. If no water is drawn the water heater will cycle every four hours and on electric will draw power for 15 minutes. That's six (needless) cycles per 24 hours. The electric heater draws 1200 watts so that is 1.8 kwh or about 6200 btu's. Assuming 80% efficiency for the burner bumps that up to about 7800 btus.
There are about 21,000 btus in a pound of propane. so it will burn 1/3 pound per day x 30 days = 10 pounds per month (for doing NOTHING).
I cycle mine in the morning. There is enough hot water left over for hand washing and dish washing until the next morning.