How could using an ungrounded appliance with a grounded extension cord be a fire hazard?
The Fire Dept is wrong - it is perfectly normal to plug a device with a 2-pin plug into a 3-hole socket.

Breaking the ground pin off a 3-pin plug, then plugging that into a 2-hole or 3-hole socket may produce an electrical hazard - possibility of a shock.

If a high-current load, like an electric heater, was plugged into that burned outlet, and the contacts made poor contact, that would cause the overheating and resulting fire, whether the heater had a 2 or 3 pin plug.