I posted this in another thread but thought it might deserve a posting of its own where it may be more widely seen, since it seems to be a persistent myth.
Ripe tomato fruit is not toxic to dogs, though for some it could cause some tummy upset. For those, of course, don't feed them tomatoes. And even ripe some websites recommend limiting the consumption, though I suspect that's more about potential tummy upset than it is about true, dangerous toxicity.
Leaves and stems however are toxic, so make sure your dog doesn't graze on the green parts or on unripe green tomatoes (as opposed to ripe green tomatoes. Yes, it is a thing.).
My guess is that this is similar to the misunderstanding that dogs can't eat avocados, which like tomatoes could cause some stomach upset, but then a lot of fruits and vegetables can cause this in dogs, and is less about toxicity and more about digestibility. Be careful of the pit though; a dog could choke on it if s/he tries to swallow it whole.