Pilot lights fine, furnace fires up, when it turns off the pilot goes out

So are you saying the pilot stays lit, but the main burner won't always fire. If this is a millivolt system you could have a weak pilot flame, not getting the pilot generator hot enough or you could have a weak pilot generator. If you are handy you can try cleaning the pilot. It may be better for a qualified service tech.

Pretty much the only thing that will do that is the thermocouple not generating enough voltage for the valve to say it's OK.
Usually you need to hold the button for pilot lighting for 60 seconds. You can use a meter to check that it's generating enough power. The thermocouple has to be in the pilot flame properly.

Check this video:
https://www.youtube.com/watch?v=suen7QlJl4o

Gas furnace will not light Thermocouple test with the multimeter

Or see this site:
How to Test Thermocouple with Multi Meter Heating and Cooling

Mots the t stat or gas valve check connections first most likely it's the gas valve cause the furnace is old valve is stuck cannot open

make sure wall is between 5" and no more than 9", adjust gas pressure or have the plumber call cozy tech support at 502-589-5380

Try jumping the two thermostat wires together to see if the heater starts. If so, replace your thermostat. If not, jump the two terminals where the thermosta twires attach to the furnace. If it starts, you have a bad thermostat wire or connection.

if you are by passing stat at the gas valve and it is still not working,i would assume a bad control valve.

try changing the thermocouple may be worn out also check the flue heat limit switch

Best regards
Greg

The pilot orifice could be dirty causing the pilot flame to be too small. Unscrew the pilot tube at the pilot assembly. The orifice is inside. Remove the orifice and blow it out with compressed air only. Reassemble and see if that fixes the problem. If the pilot flame is set too low, sometimes opening a door will cause a draft to blow out the pilot. If the thermocouple is good and the pilot flame is set properly, the control valve may be bad.

Make sure all the wire connections are good and tight. Next lift off the cover and look for a little thermal disc mounted up by top of the heat exchanger. Most have a re-set button but not all. If you have this ,by-pass it ,see if that solves the problem, if so it may be defective, or doing its job and you have a venting problem. If you don't have this and have changed the thermocouple (was it a copper tube with a fitting that screwed into the gas valve, or did it have 2 wires, that is a pilot generator, kind of same thing) make sure the probe is set correct, with the pilot flame heating the top 1/3rd of the probe. The only thing left is the gas valve.