If you have another heat source, like a wood-burning stove, you’re better off using that until the power comes back on. And if the situation is really dire enough that you have to use your car inverter to run heaters in your home, you’re probably going to be better off using that gas to get yourself to an emergency shelter or warming station.
Car power inverters are great, but they’re designed to be used when the engine is running. When the engine isn’t running, the inverter draws stored power from the battery rather than relying on the power generation of the alternator. Since car batteries have a finite amount of power storage, using an inverter when the engine is off drains a battery quickly. In fact, a typical car battery will have less than two hours of reserve capacity, which is defined as the amount of time that the battery can power a 20A load before the voltage drops below 10.5V. Letting the charge drop that low, or lower, isn’t very good for the longevity of a battery, which is why it’s bad to let batteries die.
If you plug an extension cord into the inverter in your car and use it to run electronics in your house with the engine off, you may find that you can’t start your car later on. For this reason, recreational vehicles and other automobiles that have to deliver a lot of power when the engine isn’t running typically have one or more deep-cycle batteries devoted to that purpose.