How safe is an unvented propane heater?

They burn very clean, no byproducts. Propane is not an irritant, does not cause sensitization, and has no known teratogenic or mutagenic effects. If you don't have a propane tank outside that is your biggest concern. Building codes and commonsense advise you to keep propane cylinders outside in case a cylinder leak occurs, which you may not notice since propane is heavier than air and collects at floor level.