Hydrogen peroxide administered at the appropriate concentration by a properly qualified and trained dental professional does not damage the teeth.
Some hydrogen peroxide will diffuse into the pulp and cause transient irritation of the nerves [1]. This may cause sensitivity. However, it can go away once the bleaching process is stopped.
Over the counter (OTC) dental bleaching products are only allowed to contain up to 0.1% hydrogen peroxide. On the other hand, prescription dental bleaching products can contain up to 6% hydrogen peroxide. Despite that, the risk of serious mucosal irritation or tooth sensitivity from OTC bleaching products is extremely low.
1. PubChem. Hydrogen peroxide. Nih.gov. Published 2019. Accessed November 19, 2019.