Check the high limit switch. If the furnace is allowed to continuously over heat, It's likely the high limit switch has burned out.
Make sure that the outlet vents are clean, and clear of debris. You need good air flow both in, and out of the system. also make sure the heat exchanger is clean, so it can transfer heat properly to the passing air.
If the thermostat has an option for Fan, switch it to Fan and make sure the fan starts. If the fan does not turn on, you should contact a trained HVAC tech. Capacitors can be deadly if not handled properly. You could accidentally discharge the capacitor into yourself, leading to electrocution. Or short the capacitor, leading to an exploding capacitor. In either case, it could be deadly.
Fixed! Turned out, it was an issue with the condensate drain line.
I had moved it and it had become just ever so slightly raised, so that the line had become double trapped. The collector was filling up with water, and tripping the secondary pressure switch.
The reason I wasn't getting a code, was because the water was gurgling, or bubbling, as the negative pressure pulled the water back up the condensate line. This caused the pressure switch to fluctuate rapidly, which is also what caused the sputtering flame problem. This rapid fluctuation, which was picked up by a multimeter, was too fast for the control board to actually register a fault.
Fixed the condensate line, and all is well.
Haven't been able to explain the cover on/cover off phenomenon, may have simply been coincidence. It ran all night, all buttoned up, without issue.
Best Answer
Have you checked the temperature inside the furnace? It is possible the pilot light went out.
Disclaimer: I don't hold any special knowledge about furnaces. They are dangerous if tampered with. Re-lighting pilot lights can blow your house up. I don't recommend anything.