Also, from what I've heard, hand sanitizers aren't great for you. Placing them on doors, encouraging (or perhaps pressuring) people to use them frequently, might do more harm than good. It would be vastly more practical and effective to strategically place motion-triggered sanitizer dispensers after doors, especially in patient rooms and on the way out of certain sections of the hospital.
It's respectable that you're thinking outside the box and looking to solve a real problem, but this needs more thought.