Various (Googled) 'mean time to failure' tests done on electronic gear shows that the number one failure is at turn on. The failure IS from the sudden inrush of current, and the sudden change in teperature). This failure shows up the most in comparisons of time of use... hours of use.
Say a component has a mean time to failure of 100,000 hours, The number one cause of that failure happens at turn on. (just like filament light bulbs, usually they burn out when turned on, rather than while in use)
So if you play your stereo once a week, turn it off.
If you play it every day.. leave it on.
As usual, power amps with tubes are a different story, and should be turn off. (due to possible hazards of tubes blowing up, and fire. and the expense if retubing)
So the answer is no, the heat generated by the normal use is not going to shorten the lifespan of an electronic device being left on 24/7 If it is used anyway a few times a week. In fact the lifespan is shortened by turning it on and off a lot.
So the tradeoff is only when the item is left on, but not used for long periods of time.
I would just toss out a guess that the balance is around twice a week. (pure speculation...) So if you play your stereo more than twice a week. leave it on (if the electrical energy use does not bother you)
If you use it more than twice a week, then it is better being left on. (again, aside from wasting electricity.
And if like me, you use it every day, for many hours, leaving it on is a no problem good idea.
If you use it only occasionally, (once a week or less) turn it off between uses will maximize the lifespan.
This is my understanding of the tradeoff between hours of use, vs the stress of turn on/off cycles.
(And note Bryston tests the amps with a turn on, turn off series to stress them the most they can, and not just turned on for a long time!)
Added: the heat issue: the normal heat from a component is not a factor.
Heat IS a factor if it is substantially higher than the normal temp the unit is designed for. So sticking you Bryston amps in a small cabinet with no ventilation is not a good idea. But for normal use heat is not an issue.