Why is our water heater running the burner for so long and why is the relief valve running?
"the burner had been running... even after we had opened the faucet"


Every time you open a faucet on the system you introduce cold water into the heater, necessitating additional burner operation to bring the water to the temperature setpoint. The longer you run the hot water, the colder the tank water becomes and the longer the burner will operate to heat it up.

Also, older water heater T&amp;P valves are notorious for dripping/leaking. You may need to replace the valve, it's easy.

The fact that you have had no noticeable pressure increase and that your water temperature is normal seems to indicate a properly functioning thermostat and gas valve. Test the temperature then turn the thermostat down and see if there is a corresponding response.