Could using a peroxide toothpaste cause my mouth is sore?

It Might: Some people's mucous membranes are sensitive to peroxide in some toothpastes. If your goal in using a peroxide-containing toothpaste is to whiten the teeth, then discuss other tooth whitening options with your dentist. If it is for treating a gum problem, that should be discussed with your dentist, as the toothpaste you use should be part of a total oral hygiene approach to treat your gum problems.

Maybe: There are very few peroxide containing toothpastes but if you abrade the gum with your brush it could irritate.

No: If it is an approved "ada" toothpaste then no.