Do hamsters carry diseases that are harmful to humans?

Rare but can happen. Hamsters can directly or indirectly transmit Rabies, Tularemia, Salmonellosis, Lymphocytic choriomeningitis virus, Dermatophytosis (ringworm, dermatomycosis), Rodentolepiasis and Hymenolepiasis (tapeworms).
Rare, but possible. Any household pet can potentially pass along a disease to humans (zoonoses). However, hamsters are usually relatively benign in this regard; the biggest concern from hamsters regards bites and allergies to their dander. Guinea pigs are known to be carriers of the causative agent of salmonella sometimes. It is always a good idea to wash one's hands after handling a pet (including dogs).