Why is it dangerous to burn carbon based fuel like methane in a closed room?

Two reasons. Technically if the room is well sealed, the oxygen level in the room could fall to dangerous levels, which could lead to death. The oxidation process (burning) will remove oxygen from the room. Proof is the old drop a lighted match in the bottle, close it up with your finger or lid, and it will go out. Using up the oxygen and also lower the air-pressure in the bottle.

Second reason is incomplete burning which always happens will produce CO which is carbon monoxide. This easily enters the blood stream via lungs, and will attach itself to a site on red blood cells, hemoglobin. It is not easily given up in the lungs and accumulation of CO will give you that typical "blueish" finger nail or poor skin color, and you will pass-out. If you are not rescued, in time you will die from lack of oxygen or carbon monoxide poisoning.

In addition, if there is any sulfur compounds in the fuel as impurities, additional toxic materials will be present and can add damage to the living cells, and of course cause death.