Can I leave my RV water heater ON without water pressure hooked up? This question has been posed to us. I suppose you could, but our answer is NO. There is no reason to leave the water heater on while the water is shut off.
If you want warm water soon, then I will leave the water on. There are reasons not to leave the RV water heater ON all the time.
Suppose the water heater has more air than water in it. In that case, the air will expand more than the tank might be able to handle the expansion factor of air being higher in expansion than water.
The water in the water heater is a closed system until you turn on the faucet, and then no air enters unless it is bubbles in from your supply line. Those typically bubble back out through the faucet, and you can hear it sputter.
Again the hot water tank is assured of being full. If you shut off the supply for any reason, there is a chance of air getting in the system without the water pressure supply. This might not happen but poses a risk without water pressure on the inlet side.
Any dripping hot water faucet or dripping supply line might cause air to come into the system. This could result in a water tank bursting.
In winter, leave water pressure on and do not shut off the water heater, ever, because it keeps that tank from freezing. We have a feature on winterizing your RV if that is the reason to leave it on.
In summary, I leave my water supply line on. When I leave for an extended period of time, I shut the RV water heater off when I shut the water off. Just make sure You do as a procedure to protect the hot water tank.