It really depends upon the actual incident. In general it's difficult to hold the New York City Fire Department (FDNY) liable for damages because so many traffic laws simply do not apply to emergency responders.
If you were found at-fault for the accident, you won't be able to seek any type of recourse from the FDNY's auto insurance and will be responsible for your car's own damages.
For instance, if you saw and heard the lights and sirens of the FDNY vehicle and failed to yield the right of way (required per Section 1144 of the New York Vehicles and Traffic Law), then you will be found at-fault and liable for not only your own car but also damages that others sustained.
If you were illegally parked (e.g., you parked in front of the fire hydrant) or otherwise are found responsible for your vehicle being hit by the FDNY, then no one is going to pay you for damages to your vehicle. The only way that you will be paid for your loss, if you are at-fault, is if you had collision coverage.
Now, if you are a victim in this accident, meaning that your car was legally parked or stopped, and the damages to your vehicle were due to negligence by the FDNY, the FDNY may be liable.
State and government vehicles are required to carry coverage for incidents that they are at-fault for, and if you have cause you can file a claim with their insurer. To make such an auto insurance claim, you will need to provide proof that the FDNY was at-fault.
Finding fault with the FDNY may be difficult. Under Section 1104 of the New York State Vehicle and Traffic Law, emergency vehicles such as fire engines, ambulances and police cars are permitted to go through traffic lights, and stop and yield signs, go the wrong way on on-way streets, turn in directions not normally allowed and use excessive speed if they are responding to an emergency.
Although allowed to drive against the normal "rules of the road" when responding to an emergency, vehicle drivers are required to exercise due care. So, if your car were demolished due to the act of the FDNY driver, you would need to provide prove that the driver acted recklessly, or intentionally damaged your car, in order to prevail in your car insurance claim or lawsuit if it came to that.