Will removing my son's tonsils improve his allergies?

No. Removing the tonsils won't change your sons allergies. However, they might help some of the symptoms. If your son snores badly and he is allergic, sometimes his snoring and airway symptoms might be in part due to allergies and in part due to his tonsils. Removing the tonsils might therefore be thought to help his "allergies.".
No. Tonsil surgery is helpful in the treatment of recurrent tonsil/throat strep infections, recurrent peritonsillar abscess, sleep disordered breathing, sleep apnea due to large tonsils. It has not helpful in treatment of allergic disease.
Yes, indirectly. A child who has big tonsils and big adenoids, with bad allergy symptoms, likely has trouble breathing through the nose and at night has trouble sleeping properly due to snoring. Taking the tonsils and adenoids out, along with using steroid nasal sprays in the nose, or radio-frequency treatments to shrink nasal blockage, may make his life a great deal better (allergies won't block his breathing).