Pet food manufacturers formulate cat food to meet the nutritional needs of cats, so feeding a dog a consistent diet of cat food will lead to deficiencies and possible health problems. The occasional cat food treat will not harm a dog, nor will it become ill if it accidentally eats cat food.
Cat food is high in protein and fats, and the amount of protein in cat food is too high for most dogs to digest properly. Feeding a dog a diet of cat food will eventually lead to problems with digestion, including vomiting and diarrhea.
Cat treats are perfectly fine to feed a dog or use as incentive rewards during training. However, any dog that vomits or seems to have gastrointestinal problems afterward should avoid eating any type of cat food.