Christmas is not a big celebration in Greece, at least not as big as in other countries with a large Christian population. Easter is the major holiday in Greece. I mention this so that you know that things will not be closed extra days because it is Christmas week. Your biggest issue will be that you are visiting during off-season.

Here in Cyprus, in certain places, tourist facilities are essentially shut down from November through April. This may be the case in places like Santorini.