Assuming your forced air gas furnace and ductwork is/was properly installed shutting vents completely off can actually cause you to use more fuel and possibly cause damage to the heat exchangers of your furnace. Your furnace is very dependant upon the proper airflow to function properly and efficiently. By shutting off registers you can actually decrease the airflow across the heat exchangers causing it to run less efficiently. The burners will actually cycle off of the high limit switch with the fan continuing to run. The short cycles can cause moisture to build up where its not supposed to in the primary heat exchangers causing them to rust prematurely. The heat build-up from the furnace cycling on the high limit can cause the heat exchangers to bust open causing a very unsafe situation for you as it allows carbon monoxide to escape into the supply air thus come out the registers you have open. On the newer furnaces with negative pressure heat exchangers when the cracks in the heat exchangers become to large your furnace will cease to operate because the induced draft blower cannot achieve enough of a vacuum on the heat exchangers to pull in the vacuum switch possibly leaving you without heat during the coldest part of the winter.
Hoprfully your landlord is required by local ordinances to have the furnace serviced yearly and a CO test performed during this service. If not then perhaps you should talk with your landlord and suggest they have a qualified heating/cooling contractor inspect and service the furnace. Express your concerns about the high heating bills and ask they have it serviced to insure it is operating at peak performance.
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