When you tell it to, you usually have good control around what kind of thing you want it to consider a problem, how much effort you want it to spend trying to prove that, and what should happen with code it can't prove that property for. Most commonly, people just use the "tell me if I've made an obviously dumb infinite loop which does nothing useful while it runs".

E.g. y-cruncher does something other than infinitely loop for no reason so it wouldn't trigger the loop detection logic on most anyone's compilation settings.

On the other hand, a loop which does nothing but run until the last digit of pi would either trigger the compiler to say this is a useless infinite loop or to say it isn't able to prove this loop stops (almost certainly the latter, as such a loop is a great example of going out of your way to fool the compiler).

A loop which sometimes terminates and sometimes doesn't depending on runtime input would depend on your compiler settings, but almost always nobody sets it in a way to say that's a problem (and when they do almost always it's not possible for the compiler to prove one way or the other) if their compiler even supports that kind of analysis.