How can I stop my fireplace from disabling my thermostat?
You've got a few options:


Stop burning fires. Even without throwing off the thermostat, they tend to make the house colder, not warmer, because all the heat is sucked up the flue. The exception to this is a high efficiency wood stove.
Turn up the thermostat to compensate. You'll end up constantly adjusting it as the fire dies down.
Replace your thermostat with a wireless sensor. This is wired either at the furnace or where your existing thermostat is located, and can be easily undone when you move out. The level of effort is no different than replacing your thermostat (several wires to attach and not accidentally lose in the wall).
Move the thermostat. Since your renting, this can be difficult, but the landlord may be willing to do this as gregmac and bcworkz mention below. If all you're doing is moving the thermostat to the other side of the wall, this is an easy change.