Why should or shouldn't I peel button mushrooms?
The only motivation I'm aware of for peeling button mushrooms other than wanting a very clean, white appearance is to remove dirt, dust, grit, compost, etc. from the surface without the trouble of brushing or washing. It's arguably quicker to peel mushrooms than to brush them (though not in my experience) and many people don't like washing them because they absorb a little bit of the water, which is counter-productive when e.g. sauteing. However, peeling them means you are wasting some of your ingredients, and even a very dirty button mushroom is trivial to clean by washing or brushing.

I rarely find it necessary to clean supermarket mushrooms at all as they tend to be very clean on the shelf. If they're dusty, a gentle pat with a clean, damp sponge does the trick. I would advise against peeling unless you're going for a specific visual presentation that requires it.

See also: How to clean mushrooms?