Yes, tortoises do bite. While they are typically very docile and peaceful, they are able to use the strength of their beaks to bite each other, other pets and sometimes humans for a variety of reasons.
Tortoises dig, bury and burrow instinctively to regulate their temperature, prepare for brumation and to escape predators. The desert tortoise, for example, digs itself into the ground to escape the burning desert sun. Some breeds dig elaborate burrows or dens for communal brumation. Others use it as a strategy for escaping predators.