Have you spoken to the pet owner directly? For all you know s/he could be sympathetic. Support your case by getting an allergy test. Your doctor can order bloodwork that will determine whether you are allergic to dogs. I had this done - there are excellent, accurate tests for animal allergies, and the documentation will help your case tremendously.
If the owner is unsympathetic, and you think you will not be putting your job in jeopardy, then write a letter to your supervisor, so that you're beginning a paper trail. Keep a photocopy of your signed letter. If there is no change, you can write a follow-up letter and copy your supervisor's supervisor.
Finally, there are also excellent medicines to help against allergies, and your doctor can pinpoint those for animal dander. I'm fairly sure there are over-the-counter medicines that will not cost a lot. In the end you may opt instead to keep your job out of jeopardy, maintain healthy relationships with your co-workers by avoiding antagonizing them, and address your health concerns effectively all at the same time.