Some wonderful entries here.
I second the mention of the Laurentian Library in Florence.
Was just there over winter holiday. It is truly awe-inspiring. Not for use of course, but quite an experience.
I haven’t seen a mention here of the Portuguese Reading Room in Rio de Janeiro. Truly beautiful.
Of course there is the Bibliothèque Nationale de France – NOT the useful but hideous (on the inside) François Mitterand site – but the Richelieu site, with its wonderful reading room designed by Henri Labrouste. (I don’t think they allow visitors though.)
Also the Bibliothèque Mazarine in Paris. Complete with creaking wooden floors and smooth-eyed marble busts staring down at you…
Finally, must add, above all, though it’s obvious, and this entire forum begins with a picture of it, that I have left my heart, more than once, in the Rose Reading Room of the NYPL. No place like it in all the world.