It's not that the mirror or box is damaged by heat, it's that each bit of heat energy comes from a bit of light energy. Eventually the light bounces enough times that there's no energy left in it.
A few notes: * There's no such thing as "absolute hot" state that meansno more heat can be added * Blackbody radiation means that above a certain temperature, regardless of what you make your mirror out of, it will be spontaneously emitting visible light at all times.
https://de.wikipedia.org/wiki/Perpetuum_mobile
Same principle.
Basically, what you propose negates the nature of reality. There is always friction/energy loss into heat (increased chaotic movement). Only way to deal with it, if you want permanent cycles, is constantly add energy in the same amount that is lost.