My knee jerk reaction to the question of what food to banish from school cafeterias is peanut butter. My son is deathly allergic to it, along with several other foods.
When I stop and take a moment to think, my answer is far different. Except for very young children in daycare or preschool (possibly Kindergarten as well) who have snacks during the school day and have a great deal of physical contact with each other, I don't believe that food bans are a benefit. Unless a child who is vulnerable to airborne protein particles is present, I would not ban any food from the typical school.
Instead I strongly believe that all food should be confined to the cafeteria, and that students should wash their hands upon leaving. Food should not be in classrooms, period. Not as treats, not as snacks, not as rewards, not as manipulatives, not for any reason. There are far too many dangers. Stringing macaroni on yarn endangers the child with celiac disease. Cheerios just introduced a peanut butter flavored cereal identical to the original, which could be potentially deadly if used for counting. The list goes on.
Don't ban foods, just keep it in the cafeteria.