Happen to have a cold air damper on your setup? I have an old furnace and have a damper that automatically opens when call for heat comes in....and if it gets stucks and doesn't open, my furnace never turns on. It was being sticky for a while (think worn gears in the motor)....and had the exact same issue you describing. Now it's basically always open and furnace always works. Some day I will get around to fixing it properly, or not....as I will be moving to a new place.
Those furnaces have panels that just pop off...nothing scary about that if you wanted to look on the circuit board. None of that has anything to do with actual gas line, so nothing scary
This is what a damper I have looks like: