I think you mean the converter/charger was on. It changes 120 volts to 12 volts to charge the battery and supply 12 volt
power. An inverter is quite a different thing, it takes 12 volt
power from the battery and supplies 120 volts to special outlets (in the factory version anyway). It would not supply anything to the water heater.
Back to your question. If neither the gas or electric switches were on, no damage occurs with a dry tank. If the electric switch was on with no water in the tank, you will quickly burn out the electric heating element. If the propane switch was on, usually the thermostat in the water heater itself will keep the temperature down to the point that no damage will occur (again, no water in the tank). That however is a bit hard to say for sure. I have done it once by accident with no issues in the long run. The burner will run for a very brief time, then shut off. Then repeat. Not good for the unit but probably will not hurt it.
I have no idea what the noise you are hearing is (or was).