Museum libraries and other specialized libraries/archives will often open their doors to those with an articulated research need. In most cases, you must make an appointment beforehand.
How can you find out these libraries exist? Well, most museum websites have a well-hidden library page with info. Or you can try searching the website. Another trick is to search Worldcat.org, which collects many library catalogs in one place. If you're searching for a book, it uses your location to tell you where the nearest copies of that item are found. If you're in New York looking for an exhibition catalogue, you will likely see it at the Whitney, the Met, and MoMA as well as local colleges and public libraries.