What sort of heater? If it is the circular type it will still have hot water circulating through it when the fan is off.
The later heater should be cooled by the fresh air trunking
The later heater doesn't appear to be cooled by the fresh air intake. It still gets pretty warm when the water tap is open.
You say the heater 'isn't on'. Do you mean the fan, or have you actually closed off the water tap too? If you haven't turned off the water tap, then hot water will still be flowing through. If you have turned off the tap... odds are it isn't closing properly!
I leave my heater tap open all year round anyway, as it gives a bit more engine cooling, and helps water to circulated from the back of the head. I just drive with the window open to keep cool, and avoid touching the hot heater!
Use it as it was designed to be used, turning the water on once in a while won't hurt particularly on a cool evening.
As has been said elsewhere the cooling system is over specification for the engine size.
If you are stuck in traffic for a while and the engine shows signs of misfiring, turn it on and see if it helps.
"Once you break something you will see how it was put together"
It does seem to be possible, albeit rare;
Thermostat Failure
The thermostat is pretty simple and rarely causes problems, but when it fails the results can be disastrous. The worst case scenario is when the thermostat sticks shut, which can happen if the wax element has been damaged by previous overheating, corrosion or age. If it sticks shut, it will block the circulation of coolant between the engine and radiator causing the engine to overheat.
If the thermostat fails to close, which can happen if the sensing element binds up, the return spring breaks or a piece of rust or debris jams it open, the constant flow of coolant through the thermostat will prevent the engine from reaching normal operating temperature. This can cause poor driveability in cold weather, a sharp increase in fuel consumption, little or no heater output, and accelerated blowby and ring and cylinder wear.
"Once you break something you will see how it was put together"