Why do space heaters make the air smell like that?
When air contacts a sufficiently hot object, dust in the air burns, and in some cases, particles from the hot object are also ejected similar to how dry ice sublimates.  Dust can also gather on the heating filament when not in use, and burn off immediately when plugged in.

An infrared lamp with a sealed glass bulb contains the hottest point (the filament of the bulb) within a sealed glass enclosure, eliminating filament loss.  Radiant heaters are also the least likely to stir currents of air over he heater.

For those two reasons, an infrared heater is least likely to produce a noticeable smell.