Mountain and national park trails across Italy, including the popular Dolomites region, are genuinely dog-friendly territory and heavily used by local hikers with pets — but leash rules apply, and it's worth checking the specific park's regulations before setting out, since some protected nature reserves restrict dogs entirely to protect local wildlife populations. Closer to major cities, hillside paths and waterfront promenades tend to be the most reliable spot for a long walk without needing to plan a full mountain excursion.
A leash is generally sufficient on open trails, but a muzzle is worth carrying regardless — it's recommended, and on some public transport connections up into the mountains (cable cars in particular), it can be mandatory. If a hiking route includes an overnight stop at a mountain hut (rifugio), check that specific hut's pet policy ahead of time.