The answer is -- it depends. Different people with allergies or asthma respond in different ways. It is known that increasing one's exposure to something that one is allergic to can increase one's sensitivity *overall* to various allergens. Sometimes people do seem to adjust to whatever is making them allergic and the symptoms decline or vanish. Others find that though a doctor may have told them they are allergic to a cat or dog, that actually, they don't have much reaction to them.
It can help to keep the animal out of the bedroom where the allergic/asthmatic person sleeps *at all times*, keep the home thoroughly vacuumed and dusted, use air purifiers with hepa filters (and hepa filters on air conditioning/heating vents, and a hepa filter vacuum). Also carpets will make the problem worse as they hold allergens like dander and dust. Wood or tile or linoleum floors are preferable.
Doctors say that despite common belief, there's no such thing as a breed of dog or cat that doesn't cause allergies. It may be true that some produce *fewer* allergens but ANY dog or cat will likely have an affect on someone with allergies or asthma if this is something they react to. This is because the hair of a dog or cat is not what people are allergic to. People react to the dander (small skin particles shed daily), and to proteins/enzymes in animal saliva and urine, NOT hair. So a low-shed breed like a poodle or a sphinx (hairless cat breed) will not remove the cause of the reaction. As dogs and cats groom themselves with their tongues you will come in contact with their saliva in this way even if you don't let them lick you.
For more info, try this link; there are several other topics of a similar nature on the site too.
Another good article:
Here's an excerpt, but there's lots more on how to live with a pet even if you are a family member has allergies:
All cats and dogs are allergenic (allergy-causing) to people who are allergic to animals. Cats tend to be more allergenic than dogs for allergic people, although some people are more sensitive to dogs than cats. Contrary to popular belief, there are no "non-allergenic" breeds of dogs or cats; even hairless breeds may be highly allergenic.
Dogs with soft, constantly-growing hair—the Poodle or the Bichon Frise, for example—may be less irritating to some individuals, although this may be because they are bathed and groomed more frequently. One dog or cat of a particular breed may be more irritating to an individual allergy sufferer than another animal of that same breed.
The source of irritation to pet-allergic humans? Glands in the animal's skin secrete tiny allergy-triggering proteins, called allergens, that linger in the animal's fur but also float easily in the air. Allergens are present in the animal's saliva and urine, too, and may become airborne when saliva dries on the fur. The severity of reaction to these allergens varies from one person to the next, ranging from mild sniffling and sneezing to life-threatening asthma, and can be complicated by simultaneous allergies to other irritants in the environment.