Syriacologist Steven Ring tells me that he has asked the Vatican Library whether he needs their permission to quote from their manuscripts, or produce an edition of a text contained in one. They responded:
We are pleased to inform you that you don’t need permission from the Vatican Library to quote a BAV siriac ms. on a web page.
I don’t think that any kind of copyright could apply, in fact. But well done to the Biblioteca Apostolica for saying so.