What kind of power inverter will run a 1500 watt heater using a car battery as the power source?

A heater is tungsten, which means it draws a lot more than rated current when it is turned on. So even though the heater is rated 1500 watts, it might draw as much as 15,000 watts for a few seconds every time it turns on and that will damage your inverter. Some inverters are designed to cope with that, but you have to be sure you get one of that kind.