On Christmas day I wouldn't count on it. There's not anything open in our town on Christmas Day that I'm aware of so I wouldn't expect much different anywhere else, including Branson.
As for the rest of the week, there are enough chain restaurants in town that I would expect some of them to be open as they would in any other town. I'm sure some will not be open at all on Christmas Eve day and I would expect most, if not all, to be closed on Christmas Eve itself (around 4 or 5 PM closing time).
I'm sure a lot of the mom and pop restaurants will be closed so that they can be with family themselves.