You would need a battery in order to run the heat lamp 24/7, and it will cost you at least a couple of hundred dollars, to say the least. Heat lamps consume A LOT of energy, so in reality the battery you need is probably going to cost you a thousand or more in order to last all the way through the night.
With that said, a good battery can last you 5 years or more. You can get a second hand battery for a fraction of that amount, but second hand battery may have a much shorter lifespan.
An efficient air conditioner that also does heating would require less energy and thus, less expensive battery.
Of course, there is the alternative of wood stove if your area allows it, or if you know how, use solar panel to heat water in a hot water system and use the hot water to warm the house. In theory, it is much cheap because there are place where you can get second hand hot water system for free, but in reality it is much complicated to implement such a system.
