Sometimes — with care
Ripe red tomatoes are fine in small amounts. The green parts — stems, leaves and unripe tomatoes — contain solanine and should stay off the menu.
A ripe tomato is safe and even nutritious for dogs (lycopene, vitamins A and C). The tomato plant is the problem: stems, leaves and green unripe fruit contain solanine and tomatine, which in quantity cause stomach upset, weakness and tremors. Dogs with garden access have been poisoned by grazing on tomato plants rather than by eating ripe fruit. Tomato sauces are a separate no — they typically contain onion, garlic and lots of salt.
A few cherry tomatoes or some ripe tomato chunks now and then are fine for most dogs. Skip ketchup and pasta sauce. If you grow tomatoes, fence the plants — a determined vegetable-patch raider can eat enough greenery to matter.
This article is general information, not veterinary advice. If your pet has eaten something potentially harmful or shows symptoms, contact your vet or an emergency clinic immediately. Full disclaimer.