You need 50 cubic feet in the room per 1000 Btu/h, which comes to 250 square feet with an 8' ceiling and 40k BTU. I think a converted 2-car garage will hit that; I don't think a one-car garage will. If you can hit that, you can keep this setup. (Assuming the bi-fold doors have slats to allow for airflow through the door; top and bottom of the door isn't good enough for input air.) It's using room air, but as long as the exhaust is working properly, there's no CO exhaust into the room, and it's not using up all the oxygen. (But put a CO detector in the room, just in case.)
However, this is not allowed in bedrooms, because problems at night are much less likely to be noticed until it's too late. This means this room must not be used as a bedroom, and if you sell, cannot be called a bedroom. (Also, the room is a bedroom if the inspector looks at the room and declares it a bedroom for the purpose of gas appliances.)
The alternate way of doing this is to provide an source of outside air in the closet, and replace the bi-fold doors with an exterior-style door with a threshold. This ensures there's no way to leak CO into the game room.