In our 737 simulator anyway, if the instructor triggers a fire warning, the overheat light comes on first, followed approximately five seconds later by the fire warning light and bell. After 80 knots we are taught to ignore the master caution and overheat light.
Which brings up an interesting point of view. If an overheat light comes on above 80 knots but well below V1 - the Boeing philosophy is that a caution light above 80 knots is not grounds for an abort. In this case, however, the captain is gambling that the overheat will not turn into a full blown engine fire warning. Going by what Boeing recommend he would continue the take-off.
If on the other hand he gambled wrong and the overheat light was a precursor to a fire warning, then that extra five seconds between overheat light and fire warning would place the aircraft closer to V1 (at the all engines acceleration speed).
The captain is now between a rock and the hard place where an abort at high speed due fire warning is a serious event on a runway length limiting take off - and taking an engine fire of unknown intensity into the air could be equally dangerous.
Back to simulator training (or is it testing?). An engine overheat caution light coming on say at 100 knots would, I venture to say, cause many pilots in real life to abort immediately before the situation turned into a nightmare. One does not consider whether it is a false warning or not. But this would surely be criticised as not de rigeur SOP. Far better for one's career to take the overheat into the air and risk the fire warning, and get the box ticked off, rather than risk being accused of thinking outside the square.
I wonder which any of us would do, if this actually happened on line - an engine overheat between 80 knots and say 15 knots below V1, that is. And your wife and children were down the back....