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?
cooking.stackexchange.com/users/25818/Air
I don't peel mine, or didn't until I found a "working woman" cookbook which used the stems & peels to make mushroom stock for soup the day after making grilled mushrooms.
Aside from doing that, I've never done it.
You need to be doing a lot of mushrooms to make this cost effective and I don't think mushrooms freeze well, so that's out.
It's my understanding that peeling mushrooms is like washing meat, advice previously given as a default and no longer viewd that way.
Washing meat is considered dangerous now.
I think that the mushroom thing is optional at this point, mostly I do NOT, unless I plan to make soup the next day!
cooking.stackexchange.com/users/31371/Judith Dial
I only peel my wild mushrooms if they say to peel them in the field guide, otherwise I leave them as is.
cooking.stackexchange.com/users/45469/a coder
My mother is from England, growing up they used to forage for mushrooms.
She was told to always peel them.
I found on line the probable reason; there is a folklore that if you could peel the cap - it wasn't poisonous.
BUT that is a myth!
She still wants me to peel mushrooms, but since any mushrooms I use come from the supermarket :) , I decline, and she oftens picks them out of salads.
If the mushrooms are cooked I can usually sneak them by her.
She is now 97
cooking.stackexchange.com/users/68714/J J
I`m sure it might be a good idea if eating raw mushrooms.
But I`m also sure that if cooking, all the germs and bad stuff will be killed and your mushrooms will be perfect and safe to eat.
cooking.stackexchange.com/users/77034/Trina Risucci
I peel mine, because they’re grown in manure!
As a health professional, I have an aversion to eating fecal matter, not to mention anything else that might be living in the manure, such as the nematodes mentioned below.
cooking.stackexchange.com/users/57372/user57372
I've peeled my button mushrooms since cooking with a friend once, who did it, but I've never understood why, if at all, it's advisable.  Searching shows a fairly wide range of opinions but I see no reason attached to either side.  Is it purely aesthetic?
Why should or shouldn't I peel button mushrooms?
cooking.stackexchange.com
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cooking.stackexchange.com/questions/51917
cooking.stackexchange.com/users/2167/Mark McDonald
Peeling mushrooms might be quicker than brushing them and preferable to washing due to unwanted absorption of the water, but many people do not consider it necessary to do this. Some mushroom varieties and growing methods make the skin more risky, but supermarket mushrooms are usually quite clean when purchased. Most shop bought mushrooms do not need peeling unless you are eating them raw. Wild mushrooms, and mushrooms grown in manure may need peeling. However, peeling mushrooms is generally considered an option rather than a necessity.
Peeling mushrooms is quicker than washing them and may be a good idea if you are eating them raw, they are wild mushrooms or they are grown in manure. It is not considered necessary to peel supermarket mushrooms or mushrooms you are going to cook. Peeling may be necessary for a certain aesthetic. Peeling is considered an option.