I work in a therapy office where it requires me to put hotpacks on various people every now and then. Should I be worried about getting diseases?

Always. That is why we talk about universal precautions in health care. However, the most common things you will be exposed to are the same cold and flu viruses that you would be exposed everywhere else. As far as blood borne diseases, use gloves is there is body fluids or unknown fluids present. If there are open wounds or sores, the therapist needs to look at it first.