Depends on the temperature. A furnace does eat a lot of power, so if it's running most of the time, well, your battery, even at full charge, isn't going to last that long. But if it's just running a couple of times for a short time, then look for the cause.
And causes can range pretty far. Could be a rotten battery, of course. From your description, it's doubtful that it's the solar system UNLESS your solar controller is incorrectly allowing the system to run in reverse when it's dark. (Our cheapie does that. It'll not just run down a battery, but kill it dead-dead-dead if left unattended.)