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"An investigator or monster figure in the room at the end of it's owner's turn is dealt 2 damage."
Owner refers to the fire? or the the investigator/monster? example - Investigator walks past fire - in a safe room. Ends his turn. "sweet! i take no fire damage"
I play a mythos card to "move the investigator 2 space", and i move him back into the fire.
Owners turn -
If the investigator (singular), the fire wouldn't hurt him.
If the investigators turn (right before keeper) he would take 2 burn damage
If the end of the keepers turn (owner of the fire), he would take 2 burn damage.
Also, fire REALLY does spread 1 ROOM per turn if the keeper wants it as such. on an outdoor map, this is almost impossible for the investigators to do anything.