Is there an insect bite that causes hands and feet to peel?

Probably not.: Unless there were dozens of insect bites located specifically on the hands and feet, it would be very unusual for a typical insect bite to trigger the hands and feet to peel. More likely is post-streptococcal infection or something called "hand-foot-mouth disease". Dyshidrosis is a skin condition that can cause peeling of hands and feet--typically associated with sweaty palms and feet that itch.