Many vegetables are fine for a dog to eat, and carrots make a low-calorie alternative to fattening, preservative-filled commercial treats. Dogs can eat raw or cooked carrots with no dire consequences except orange poop.
Not only are raw carrots low-calorie treats for the weight-conscious dog, they can also be helpful in helping "scrub" teeth clean as the dog chews.
Dogs are carnivores and do not have the digestive enzymes to break down raw vegetables. For your dog to get full nutritional benefit from carrots, they have to be cooked or pureed.
Use either baby carrots or chopped up large ones as treats. Carrots can also be cooked, stored in the refrigerator, and handed out as needed.
Don't worry if you see small pieces of undigested carrot or any other vegetables in your dog's poop. This doesn't indicate there is anything wrong; simply that his system can't naturally digest carrots.
Use frozen carrots as a teething air for a teething pup. The cold, crunchy texture will both ease her urge to chew and soothe irritated gums.