I am pretty sure that Montserrat is open on Christmas Day. They also do the very important Misa de Gallo there (Midnight Mass) on Christmas Eve.
As for Christmas day, here you will find a list of the restaurants that will be open:
http://www.barcelona-tourist-guide.c...staurants.html.
For an idea on what else you can expect to see in Barcelona at Christmas, have a look at this article:
http://www.welcome-to-barcelona.com/...o+Barcelona%29
Donīt forget the 26th is a public holiday in Barcelona as well, so if you need to buy any groceries, I would recommend doing so on the 24th!
Hope I helped!