A thermopile will be located in the flame of the pilot light and will not generate enough current to open the gas valve unless the bi-metal in it is at a certain temperature. If yours is not in the flame of the pilot light it is a temperature activated High Limit switch (most likely).
The High Limit switch is there to protect you and your home; please do not by-pass it. Underwriters Laboratory has demanded it there and any modification can result in Fire, Injury or Death.
A thermostat will not shut a pilot light off, only the main burner. The pilot light should be 'standing', whether there is a call for heat or not. Take the limiter off the heater and take it to a heating supply store along with the Brand, model and serial number. They will be able to help you with it.
While the store is closed today would be a good time to get out a wire brush and vacum cleaner. Clean the little holes in the burners and the large air intake holes of the burners. Remove the pilot tube and clean it thoroughly. While we are not using our heaters in the Summer there are little spiders that find refuge in the heater and build webs, cocoons and lay eggs which mess it all up. Who knows? a good cleaning may be all it needs and doesn't cost a thing when you do it yourself.
Please post back and let us know how it works out; I wish you the very best.