Hotels are open every day of the year, but not necessarily their restaurants. I would look for Chinese places, they are often open on Xmas. Li Wah, Siam Cafe, etc. You could stay downtown and easily take a cab to where these are, they're close to downtown. Not sure if Table 45 is going to be open, at the Intercontinental hotel on the Cleveland Clinic's property, but that would be another good place to ask, they might be open to capitalize on the business from visiting friends and family members at the Clinic.
Li Wah Restaurant
2999 Payne Ave Ste 102, Cleveland, OH 44114
9801 Carnegie Avenue, Cleveland, OH 44106