Heating went out one day. Thermostat would do nothing (heat/cool/auto/on/up temp/down temp). Gas fireplace, oven, dryer, fireplace all work fine. Replace thermostat, no difference. Replace thermostat with similar digital model. Replace blower filter. Open up the furnace (Arcoaire GNE model) and reset the power. Flue fan turns on. Second click and pilot ignites the burners. Excellent! Blower fan comes on, burners go out. Sometimes they relight after a few seconds, but only to go out again just after a few seconds. Cleaned off flame sensing rod. Button it up and furnace heats house to temp set on thermostat. Clicks off as it's supposed to, never comes back on with temp drop. Usually once the blower motor kicks on and the burners go out that's it. But every now and then I'll get it to stay lit and heat the house for a little while. It's a little cold and any help would be appreciated.

Cleaned off flame sensing rod
Replace it.