First remove the cover from the thermostat, if it is not a mercury bulb type, it will have two tiny contacts visible, cut a thin strip off of a business card and fit it between the contacts, and use it like a file to polish the contacts.
Make sure you get the correct millivolt rating for the thermopile... replacing it works about 90% of the time, when that fails, the gas valve needs to be replaced.
Before you replace the gas valve however look for the high temperature safety switch mounted near the top of the fire box, that may have failed.
Your only handicap here is that you do not have instruments to simulate millivolts and also simulate millivolt inputs... we used those extensively in the old days, currently I do not think any of the best service outfits even carry such instruments either.... so we are stuck with a bit of trial and error here.
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