We have a gas water heater. Sadly, it's covered in a layer of insulation that obscures all the labeling (that's probably illegal, come to think of it), so I don't know the manufacturer or the year it was installed, but if it was new to the house, it was installed in about 2003.
It has a relief valve that runs to a dedicated pipe in the ground. As of late, this pipe has been dripping a little bit, but it decided to flow some more today – not a full release on the valve, but still a significant flow nonetheless.
We opened up a hot water faucet on the same story for a few minutes, and the flow slowed to its usual level, where it remains as of this writing. Neither the water from the faucet nor the water from the valve were particulary hot for a water heater – about 130-135F. We do not own a pressure meter.
However, the burner had been running during the entire process and even after we had opened the faucet to slow the relief valve (it was running for about 1 hour 15 minutes in all). I didn't hear any water leaking from anywhere, just a uniform hiss from the burner.
The burner did eventually turn off. I don't know if 1:15 is common for a water heater to run for, and the burner not stopping doesn't seem like a common problem. Also, I just checked the heater – the burner's still off and relief valve isn't running too much, but there is a consistent hiss from the relief valve itself. A very light tap caused a few extra drops to spill out.
Best Answer
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&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.