Non allergic sensitivity turns to be a very broad term, too broad to even cover psychological situations. Nutritional sensitivities that don't cause an immunological problem are not likely to be treated by allergists or immunologists.
The "About" section in the website of the American Academy of Allergy, Asthma and Immunology (AAAAI)(1) doesn't mention anything outside the scopes of allergy medicine or immunology.
An allergist / immunologist (commonly referred to as an allergist) is a physician specially trained to diagnose, treat and manage allergies, asthma and immunologic disorders including primary immunodeficiency disorders. These conditions range from the very common to the very rare, spanning all ages and encompassing various organ systems.
In the "About" section in the website of the British Society for Allergy and Clinical Immunology (BSACI) doesn't mention anything outside the scopes of allergy medicine or immunology as well(2).
If nutritional intolerance isn't causing an allergy or an immunity problem but rather another problem, then in general, depending on what that problem is, another field of expertise would be suitable.
An allergist / immunologist might be able (or not disallowed) to try to treat such patient's intolerance but that's unlikely that he will.