I have an outdoor hot tub with the pump and heater located inside the house. As no electrical connection on the pump or heater comes within 6 feet or the tub, and is located indoors, if they are grounded, do they need a GFI circuit breaker?

Yes. The water line that connects the tub to the pump and heater is a good electrical conductor. Without a GFI, even thought the pump and heater are grounded, you won't know if there is a ground fault until something bad happens.