Holy Week in Barcelona

All museums list opening and closing hours you just have to look up each one on the web; most are always open except for xmas and New Years days.

You'll definitely want to do the online research Jazz mentioned, but if you have the opportunity to stop in at the tourist office (I used the one under Pl. Catalunya) early in your trip, you can ask for its handy-dandy summary of operating hours. There's also a list of places that have free-opening times, which can be quite a money-saver in a city were it seems as if walking in the door often costs more than 20 euros per person. Just don't try to go somewhere like the Picasso Museum during a free entry period! Definitely inquire of the T.O. personnel about days/hours specific to the Easter period, because the printed materials may contain non-holiday-specific schedules.

Everyone tends to focus on the modernista sights (I certainly did) with a side of the Picasso Museum, but don't forget the many other wonderful museums, most of which have no issues with crowds. Your guide book will cover at least the major ones, but possibly not all. That's where skimming through TripAdvisor's "Things To Do" list can be helpful.

A lot of the most popular sights in Barcelona appear to be non-government-operated, and I think you'll find they maintain pretty good hours even during holiday periods.

Places that, even if open, pretty much require you to buy a ticket before you walk up to the door (else long lines) include:

Camp Nou (soccer experience)
Palau de la Musica Catalana (tour required, but they are frequent)
La Sagrada Familia (decide whether you want to ascend a tower)
Parc Guell
Casa Mila/La Pedrera
Casa Batllo
Picasso Museum

I do not mean to suggest that you have to go to all those places, merely that if you plan to visit one of them, you should consider buying a ticket in advance, which will in most cases come with a fixed entry time. Putting those puzzle pieces together can be quite a challenge if you have a short stay in the city.

From what I've heard from a couple locals, there's not much going on during holy week in secular Catalonia. There may be some closures on Easter Sunday, but I'd bet there will be plenty of open sights. I was in Cordoba last year (where Semana Santa is a pretty big deal), and some of the tourist sights were open on Easter Sunday as were most restaurants. It's a big day for eating out, so make dinner reservations!