Can a business refuse to honor my registration?

First of all, if your dog is Service Animal, businesses MUST allow both of you to enter. This is true even if you have no photo ID card, Service Dog patches, or any other identifying information. Having your Photo ID Card and any other identifying patches, etc., will make entering a business question-free 99% of the time. You are responsible for the training and manners of your dog while in public, however, and if your dog is barking in a theater or restaurant or otherwise being disruptive, the owner of that establishment can legally ask you to either leave or to remove your dog from the business.