Can dog allergies cause dryness of your mouth and throat?

Probably not. Allergy to dog hair, or secretions result from a genetic, immunologic disease in which the animal secretions react with a substance (ige) in your blood or mucous membrane that results in an allergic reaction. The usual symptoms result from a release of a substance called histamine and most commonly result in sneezing, watery nasal secretions and itchy eyes. Dryness is not a common symptom.
Complex question. The assumption is that you are allergic to dogs. This is a specific genetic immune reaction in which the dog or its secretions cause an ige mediated immune response. Simply speaking there are proteins in your blood and mucous membrane that result in an allergic response. The symptoms are usually, not dryness, but increased secretions from the nose and throat. Consult an allergist.