Will the hole drilled into the furnace door cause problems? Gases etc. escaping?
There's no problem. In a healthy furnace, combustion gases are contained inside various components housed in the upper cabinet.
If you think through the airflow path of your HVAC system, there's normally a vacuum on the lower chamber of the furnace cabinet because it's part of the return duct--it contains the fan. That means that air is drawn into that space, joins all the other incoming return air, and passes through the furnace's heat exchanger and out through the main exit trunk. A tiny hole just creates a miniscule return path from the furnace room.
I've done exactly this with one of my furnaces after upgrading the circuit board to one with an indicator, so that I could see the indicator from outside. I applied clear packaging tape over the 3/4&quot; hole I made just for the heck of it. I put two pieces back to back to eliminate the exposed adhesive, then I framed that with tape on the inside of the panel. It held well. Obviously you don't want to use anything that could get drawn into the fan cage.