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.

The statement by "Barrie" is nonsense, many devices don't have a ground connection and thus only have a 2-pin mains plug. Such devices are "double insulated" and have the 2-squares logo and possibly some text like:

enter image description here

The damage of that power strip was very likely caused by a short circuit and/or overload. There is no ground/earth connection required for that to happen. Likewise it is unlikely that a ground/earth connection would have prevented that damage from happening.

Connection a 2-prong device into a 3-prong socket is OK. Properly designed 2-prong devices are isolated and don't need protective earth.

What is dangerous is plugging a 3-prong device into a 2-prong socket, or using a 2-prong extension cable with a 3-prong device. That will cut the protective earth wire on an appliance which needs it, exposing the user to electric shock upon failure inside the appliance.