I will be quick to caution, in this day and age, turbine engines with advanced fadec systems may compensate, and mask issues that you would normally see in an old fashioned motor.
The fadec will do lots of things, close bleeds, add fuel, etc to maintain normal operating paramaters. Example, our Q. Nothing exceeded normal paramaters until they started melting things off the engine, and then it was things like the fuel flow indication. The motor was, for all indicated paramaters, doing "just fine" by the fever check, the crew wouldn't have done much of anything until the indications started failing... and that was a fair amount of time after the initial fire warning.
This is why you will see 121 guys say, shut it down. Sure 90% of the time there is no problem. But lets say you get te 1% case, and you let it burn until other stuff starts to mess up( because there were no other indication). Perhaps, one of the things to take damage is the main gear that's sitting right next to the engine (and on fire). You put the plane down, gear fails and you hull loss a brand new $30 mil. plane. The cp is going to ask 2 questions. Did you run the checklist? What did it say to do? Manuals, I'd and jepps by my door please.
Much shorter is, did you run the checklist? Yes we secured it and landed singe engine. If te engine wasn't on fire, guess what? You followed SOP's, ran the appropriate checklist, and did so in a manner that complies with your training. Its now on the fleet manager, manufacturer, and mx to determine why there was a false indication and fix it, or come up with better guidance.
System knowledge is huge, but, IMHO, during a potential emergency is not the time to suddenly get creative if there is no need. Do as your trained, and make it somebody else's problem on the ground. Despite what anybody here says, I bet every one of the 121 guys here will shut down the engine if they get a fire indication in flight. Nobody wants to find a new career that badly right now.
Now, a small, light twin where you don't know if you will climb etc? I would prob. Spend some extra time making sure it was burning.
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