Can you say the fire is on/off ? No. A fire does not have a handle that one can turn.
These are OK: Put out the fire. (you cover the fire with something, or put water on the fire.) Put the kettle on the stove. Take the kettle off the stove.
1) My Oxford dictionary says put on can mean you switch something on or operate something. It gives the examples: put on the light, oven, radio, television, etc 2) This same dictionary says put something off means switch something off. Example it gives: Could you put the lights off before you leave ? 3) This