There is no law in the U.S. requiring airlines to fly therapy dogs. Therapy dogs are not service dogs. Some airlines might choose to fly them, but it is their choice, just as it is their choice whether to permit pets or SAR dogs. Also, the ADA specifically does not apply to airplanes.

Under the Air Carrier Access Act, they must permit people with disabilities, with service dogs to fly unless doing so would constitute an undue burden or if the animal is disruptive. For example, if the animal is too large to fit in the foot space allotted to the human traveler, they can require the human to either purchase a second ticket or to have the animal fly in baggage. If the animal barks, urinates, pesters others with jumping up or sniffing, or is otherwise behaving in a disruptive manner, then they may refuse to have the animal in the cabin, but should offer to fly him in baggage. Airlines are permitted to ask the owner of the dog what the dog is trained to do as a condition of allowing them on the plane.

Unfortunately, since there are so many groups issuing fake ID papers so that people can claim their pets as service dogs, real documentation of real service dogs is also becoming meaningless. The best evidence a dog is a service dog is the dog's behavior and the owner's description of what the dog does.

The Air Carrier Access Act is unlike the ADA in that it includes people with disabilities with emotional support animals as a protected class. So even though the ADA does not cover public access rights for people with emotional support animals in public accommodations, the ACAA does cover them for flying in the cabin of planes.

Off hand I can think of only one state that grants special rights to therapy dog owners, and that is only for mental health professionals who use them in their work (which accounts for very few therapy dog owners over all).