Can I put a sign up in my Mom's room that she is allergic to Banana's? or is that against HIP PA?

Does she have a personal physician? If so I would talk to him about this and if he thought posting a sign was appropriate in this situation I would ask him to discuss the problem with someone in authority at the place your mother is at. He might carry a bit more weight with those who are in charge there.

I would think if your mom wants a sign in her room stating that she is allergic to something, she would have every right to have it. If it was to be placed there by a care provider without your mom's permission, that would be a violation.

No lie, just wondering is she allergic to nuts too?

Why don't you just tell the staff and if they don't listen and that if something happens to your mom, you'll sue them ;o