Does it stop hurting them if.you warm up the iron with your body heat?
It varies. There is no consistent meaning to cold iron. In some accounts, it is just any iron, and it is simply its mundane nature that is poisonous to magical beings. Others refer to cold iron as a weapon of iron specifically, and it is human killing intent that magic creatures are weak to. Others refer to cold iron as literally iron ore or iron that has never been worked by fire, which is more about mystical uses with iron rather than iron itself.
