Training and Nutrition Stability Act of 2023 This bill excludes specific employment and training program allowances, earnings, and payments from income when determining eligibility for the Supplemental Nutrition Assistance Program (SNAP). (Under SNAP, a household must have an income below a certain level to qualify for program benefits.) Examples of programs excluded from income include Workforce Innovation and Opportunity Act (WIOA) programs, vocational rehabilitation programs, and refugee employment programs. This bill also removes the requirement that earnings from on-the-job training be considered earned income for purposes of determining SNAP eligibility.