Never had a fire warning on a Dart after shutdown, but some of the earlier Dart 6 machines would give a momentary warning during a prolonged climb in very hot, like ISA plus 25 or more, conditions. The trick was to lower the nose and get some airflow going and the warning would quickly stop.
With this knowledge in my background, I once nearly got myself and crew killed while doing some hot weather/short runway certification work on a much more powerful version of the same aircraft type. We were doing successive water meth takeoffs to measure unstick distances, followed by a split arse reversal turn to dump it back on the reciprocal direction runway, roll through, line up and go again. So it was getting pretty warm in the engine bays, and when the fire warning came on momentarily at about V2 plus 10, my initial reaction was to ignore it, especially as it did not remain on as speed increased. Fortunately, a pilot sitting down the back heard the bell, looked outside, and as luck would have it, saw flames on the inboard side of the cowl. He alerted us to the fire and we promptly did the drill and landed.
A combustion chamber had burnt through, heating up the firewire, then turned into a blowtorch and taken out the firewire and just about ate through some of the engine control linkages
in the 30 or so seconds that it took us to do the drill.
We did not observe any loss in engine power, or change in TGT, presumably because the water meth was pouring in at a great rate to compensate for the inefficient combustion chamber.
So, after that experience I will always, but always, at least shut the engine down just to get the fuel out of the equation. Whether to activate the fire bottle is really dependant on what the checklist says regarding warnings present, but the moral has to be 'if in doubt, fire the bottle'.