You can still buy "glow plug" type model airplane engines, and these also run with no power supplied to the spark plug. If there are accumulated carbon deposits in the cylinder head, the carbon can get hot and act as a glow plug, keeping the engine running, if fuel and air are available. If you can get it to run without fuel and air, now that would be something strange.

Something is miswired in your control system or two of the conductors are getting together somewhere keeping 24v to your Y terminal I would pull thermostat off the wall and disconnect then tie R and Y wire together and apart and make sure that contols cooling the R and G and see if that controls fan Here is a link that might be useful: Air Conditioning and Heating Repair Made Easy

I pretty much can do what I like with the apartment with some restrictions as my step-dad and mom own the house with an in-law apartment. I do know an electrician. I was hoping it was something like replacing the thermostat unit thinking something internal was wrong. As my boyfriend and step-dad can do small electrical work. Looks like I'll have to call my friend.

Not an expert but sounds as if while the burner is on the flue gases are going up the chimney. When the burner cuts out the chimney effect quits and the blower is producing a negative pressure on the chimney or a leaking heat exchanger pulling the exhaust gases into the system. Again no expert but a leak is indicated. I think the CO detectors will not detect until a minimum threshold. The limited amount of time to possible CO exposure after the burner cuts out means the detectors may have insufficient CO present to set off alarm.