If you stroll the French Quarter or ask your concierge to recommend one of a hundred companies in New Orleans that offer walking tours you'll find that most if not all are pet-friendly. Make sure to ask when making your reservation and bring plenty of water...lots and lots of water! Some of the walking tours are two hours long, mid day and hot, so water for you and water for the pup! It's a great way to see the French Quarter, The Garden District, and/or some of the graveyards.
Typically, pets must be well behaved, leashed, and are not allowed into any of the buildings on the tour, but they are more than welcome to walk along and listen to the tour guide and pick up what they can about New Orleans history.
Please remember to bring appropriate clean up materials.