If the running water noise continues once the tank is filled you may want to call a professional plumber to investigate the situation for you.
Tankless water heater noisy pipes.
Generally speaking there is a lower chance of hearing strange noises with a tankless water heater vs tank model.
Or you can try to run down the leak yourself by checking the following.
Other sources of noise include blocked or inadequate ventilation or a defective or improperly set pressure valve.
Inside a condensing tankless water heater the combustion process superheats the water that enters your shower sink etc.
Others reasons for a noisy tankless water heater model may be an issue with the burner the fan may be dirty or possibly a leak in the sealed combustion resulting in irregular gas combustion.
Even though a toilet only uses cold water i have seen some toilets do this.
The sensor controls the amount of gas sent to the unit.
Good luck with your project.
Another common problem that can create noise in tankless water heaters is a dirty flow sensor which controls how much gas is sent to the unit.
Water vapor is a byproduct of combustion although in this case the vapor contains elements such as methane nitrogen and oxygen.
Another standard issue that causes loud noises in tankless water heaters is a dirty flow sensor.
In that case you need to have a plumber install a check valve in the water line.
If your water heater is making noise like water is running you probably have a leak or a broken pipe.