I am allergic to cats. If I eat a bowl full of cat hair or cat dander or something, would I have a similar reaction to someone who's allergic to peanuts eating a peanut?
Unlikely. Your allergy operates on a certain immune response in your barrier tissues and fluids, such as your mucus membranes in the respiratory and ocular openings, and in your skin. I suspect your digestive system would simply digest those allergens. The hair might make you pretty unwell though. We're not designed to deal with ingesting hair or fur as some other mammals are.