What could be causing my furnace burners to fire up, then shut off due to partially open door?
There's likely a switch that senses when the door is ajar attached somewhere near the edge of the door opening. Sometimes the switch presses against a little feltlike pad that can become depressed over time. That'll release the switch and tell the furnace that the door is open when it's not. Replacing or thickening that pad may well solve your problem. Oh yes, while you're in there: cleaning the pilot, sensor, and blowing out the burner box is always a good idea.