Some places allow pets, some don't. I've noted inns that allow pets, but even so I don't recommend showing up anywhere with a pet in tow unless you've cleared it over the phone with the innkeeper ahead of time. Note that many establishments have only one or two rooms (often a cottage or room with exterior entrance) set aside for guests traveling with pets, and they won't be quite so happy to see Fido if the "pet room" is already occupied. Also, it's common for a surcharge (usually $10-$20 per pet, per night) to be added to your bill to cover the extra cleaning effort needed by housekeeping staff.
Several websites dispense tips and list animal-friendly lodgings and campgrounds: www.petswelcome.com, www.pettravel.com, and www.travelpets.com together contain thousands of property and campground listings. Also note that all Motel 6 hotel properties accept pets, though they are pretty rare in northern New England.
Keep in mind that dogs are prohibited on most hiking trails, and must be leashed at all times on federal lands administered by the National Park Service (which includes Acadia National Park in Maine). Pets are allowed to hike off-leash in the White Mountains National Forest in New Hampshire and the Green Mountain National Forest in Vermont, but you must control them by voice. No pets are allowed at any time (leashed or unleashed) at Baxter State Park in Maine. Some other Maine state parks do allow pets on a leash.
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.