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A space heater in space could only transfer its heat through radiation instead of conduction and convection, so only a small portion of its heat would escape. It would get hotter and hotter until it melts and stops working.
Essentially, that's how the ISS regulates temperatures. It has big external panels that radiate excess heat into space because there's very little matter up there to conduct heat away.
Wouldn't that depend on the type?
Some are radiant only, wouldn't those function normally? (I'm really not sure how much even a radiant relies on air flow to keep from overhearing).
I'm sure there are. I don't think I've seen any that are radiant only, but I have seen some that use a dish reflector to direct the radiant heat.
The oil-filled are radiant only.
I was actually thinking of the dish type when I wrote that, but the oil filled are radiant only, but surely induce some airflow and convection.
Interesting thought experiment.
The oil filled ones are still conductive/convective. They heat up the air that is directly in contact with them which then spreads around the room. You wouldn't feel the heat radiating out from across the room, for example, but if you touch the surface, it would be uncomfortably hot.