The incident, captured on a cell phone and posted to YouTube, shows the firetruck trying to reach the fire in a condominium building on St-Paul Street near St-Laurent Boulevard on Saturday afternoon.
More than 120 firefighters responded to the fire, smoke from which could be seen for kilometres.
In the video, one fire truck arrives at the scene, but is stuck behind police cruisers on the narrow road. The truck rear-ends the cruiser to force it forward, at the same time taking off the bumper of another unoccupied car parked on the side of the street.
The Montreal fire department said the manoeuvre was "exceptional" and the city will compensate the owner of the car.
No one was injured in the fire and the cause is still under investigation.