Dogs are broadly welcome across Italy, particularly at outdoor seating (dehors), and there's no blanket national law banning them from restaurants — the decision is left to the individual establishment, and many places welcome pets at outdoor tables even without any official stated policy. Travel writer Gigi Griffis, who has spent months exploring Italy with a dog, notes rarely if ever being turned away from a restaurant across the country. If a specific place doesn't allow pets, it will typically say so with a small sign near the entrance.
The two places dogs generally aren't brought, by custom rather than strict law, are supermarkets and churches.