Airbendane has a good point. For airline transport civil machines, the fire zone is very beefy, and can withstand 2000 degrees F for 15 minutes without affecting the basic airworthiness of the machine. Military engine fire zones are not nearly so capable. This gives some leeway for careful diagnosis, I think, and the ability to look for confirming indications. I also think the history of the type should give some guidance, since some fire detectors are sensitive at sunset, or on hot days, etc.
For a civil transport generally, to truly believe the fire warning enough to act on it, one should carefully look for some confirming secondary indications, I think. If performance is good, and the available landing area close, then a shutdown isn't so drastic a choice, so do it. But if it is a cold sea, or a long way across inhospitable territory, and all there is is the one light, a delay in shutting down that engine is reasonable. For a less protected military or Part 27 helicopter, there is less comfort in waiting, so the harsh choice might have to be taken.