NICET certified fire alarm tech for 20 years here... HVAC shut down is done in coordination with dampers not to contain the fire, but to prevent moving smoke through the buildings. smoke is the killer, not so much fire.
stairwell pressurization fans force fresh air into stairwells to provide egress routes with breathable air.
AFAIK there's no design consideration to attempt to smother a fire
smoke is what kills you in a fire, so getting the smoke out of the house as soon as possible is a good idea.
any risk of "feeding the fire" is operating under the assumption that all the oxygen has burned out of the space, so the occupants are already dead at that point.
if you want to turn on your fan, you also need to provide a means for fresh air to enter the building, or you're just going to create negative pressure in the structure and not end up moving much air.
if you can automate a window opening as well, i don't see any problems with running the house fan.