If you place your photo to be copied manually, rather than through a manual feed, you needn't be concerned.
Bright light causes chemical reactions any time the photons interact with a surface, but the brief flash of a copier, one single time is unlikely to be of any consequence.
My wife has been a photographer for over twenty-five years and has used the very similar process of scanning to convert her older photos to digital versions many times without damage to the original. In fact, scanning them or having them scanned to a digital format (jpg, etc.) may be an even better idea in terms of preserving them.
Unless you have something extraordinarily frail, you should have not problem. If your original is extremely delicate, it should be handled by an expert in such reproductions. But that did not sound like the sort of copies you were talking about.