This is a common reaction to benzoyl peroxide, its pretty harsh.
Just use very gentle products on your skin for the next few weeks and your skin should go back to normal. I didn't check to see where you're located so IDK about brands, but use a cleanser and moisturizer for sensitive skin and make sure both are non-comodogenic [won't clog pores]. Don't use scrubs if your skin starts to peel because they're too irritating, just rub very gently with a cloth and warm water to help exfoliate.
If the moisturizer is stinging for more than a few minutes I wouldn't use it. Try one thats labeled for sensitive skin if you're using normal lotion currently, or I guess you could try olive oil or something like that if all types of moisturizers are bothering your skin. In general though its fine to use moisturizer in this situation as long as it doesn't bother you too much.