Pulling the fire handle will close fuel, hydraulic, and bleed Shut Off Valves. I assume it's residual burning. The video clip I saw was only a few seconds.
I had an engine failure 15 years ago. A guide vane came off a turbine wheel and shot everything out the back of the engine. The fire warning never came on in the cockpit. Once on the ground, all we could see was a little smoke. Yes, we ran the emergency checklist for severe damage/fire, it's the exact same. It was weird pressing a fire button that never lit like it does in the SIM.