Major indoor attractions — museums, galleries, and landmark monuments such as the Colosseum or the Vatican — generally don't allow pets, with the common exception of very small pets carried in a bag, and always excepting certified service animals. This is genuinely changing in places: a growing number of open-air museums, markets, and piazza-based attractions welcome leashed pets, though this is still the exception rather than the rule and rules vary site by site.
Outdoor attractions in general tell a much more welcoming story: piazzas, public gardens, and historic town centres are almost universally fine for a leashed dog.