There are two primary types of insects whose larvae damage fabrics.
Carpet beetles (Family: Dermestidae) and Clothes moths (Family: Tineidae).
Both carpet beetles and clothes moths are unusual in the insect world because they are among very few insects that produce an enzyme in their digestive system. This enzyme, keratinase, that allows them to digest keratin, the protein in animal hair.
There are other insects that can: crickets, silverfish, cockroaches and termites, but infrequently feed on fabrics.