They don't have any obligation, but then again neither has the government any obligation to permit/not penalize the business. If Facebook wants to operate in the EU, it will have to follow EU rules.
Personally, I'd rather not operate in a country than face arbitrary censorship requests, but then, I don't run a billion dollar business (and even countries like EU and US have some fairly "arbitrary" censorship requests).
I know you're being ironic, but I think that's the way it should work. The customs and culture of a country should be respected and followed; the same we ask immigrants coming to first world countries to adapt to our customs, our businesses should adapt to theirs if they want to sell there.
And in muslim countries, if someone posts LGBT or atheist positive posts, they should immediately be moderated and the poster reported to authorities.
Facebook should also prevent users from these countries seeing such content posted from more liberal places.