The gas vent from my furnace is very hot and I think my heat is going out. I have huge gas bills and my apartment is cold.
Contact your utility, and have them do a comparison to other units in the building, and the previous owner. As for the flue, how hot are you stating that it is getting? Have you tried to talk to the property owner to see what their assessment is?
Here is a fact, you live in an apartment. You are at the mercy of who owns the building, so what ever they do with the equipment, is their decision. That means, if they do not want to maintain it, and it is faulty, talk to your local renter authority where you live, to see what your rights are.
How do you determine if the gas fired equipment is faulty. You need the correct instrumentation to take the readings for the gasses, heat, etc, to determine.