Even if the C wire was not connected, it should not be using up batteries, unless it is the radio for connecting it to the home controller, that is causing it. Do you have a C terminal on the board or thermostat connection point inside your furnace? Is there a wire connected there and at the thermostat?
It is possible that the C wire is broken, or the installer did not cut enough insulation off, or just never connected at all.
It is possible that the C wire is broken, or the installer did not cut enough insulation off, or just never connected at all.