Are organs of a celiac patient safe to donate?

Yes. Should not be a problem, as long as they had good function before. Celiac can affect function of a lot of organs.
Probably yes. The 2 organs that are usually affected by celiac disease are the intestine (damage to the lining) and the skin (a rash that looks like herpes, called dermatitis herpetiformis. Other organs should be safe.