The quick answer: Proceed with caution. If the potato is baked or boiled, it is safe to feed to dogs. But dogs should not eat raw potatoes.
Benefits of potatoes
Potatoes contain carbohydrates for energy, as well as potassium, iron, and vitamins A, B and C. Plain baked, boiled, or mashed potato (after it’s cooled) is safe to feed in moderation.
Unripe potatoes: Potatoes are a member of the nightshade family of vegetables and contain a compound called solanine which is toxic to some dogs. Unripe potatoes as well as the leaves and stem of the plant all contain this defensive poison (so keep curious chewers out of the garden if you grow potatoes). However, cooking potatoes reduces the levels of solanine and makes them safe to feed to dogs. Plus, a potato that has been boiled or baked is much softer and easier to eat!
Note: Sweet potatoes are not in the nightshade family and contain more nutrients than white potatoes.
Added seasonings and fats: Most of us don’t eat plain potatoes, and it isn’t unusual to add salt, cheese, bacon bits, and seasonings to make potatoes extra tasty and enticing. However, these additions also make potatoes less healthy. For your dog, it’s best to stick to plain potatoes since some seasonings are toxic to dogs, and too much fat (whether it’s added cheese, butter, or in the form of French fries cooked in oil) can mean trouble for your dog. Not only do the extra calories set your dog up for weight gain, but too much at once can cause vomiting, diarrhea, or pancreatic pain.
Moderation: White potatoes are high in carbohydrates—only allow your pet to eat bites of potato in moderation. For diabetic dogs, consult your veterinarian before offering potato as a treat. Added fruits and vegetables should make up no more than 10-20% of your pet’s diet.