What happens if a patient is in a state of comatose (unresponsive) can their family tell the doctors to pull the plug and let them die?

Yes and No. There are laws and protocols that must be followed. Depending on the reason for coma, the prognosis may be poor, and likely, this is your concern? I'm sorry if you are in this situation. It is not inappropriate to withdrawal care and place the patient on comfort measure, but again, their are laws and protocols that dictate who makes that decision.