Your local library contains many non-fiction books that are filled with inaccurate statements. Librarians are happy to help you with fact checking research but it wouldn't be appropriate for them to label some books as fake. Make up your own mind.

I'm imagining a situation where a book was discovered to be an unequivocal lie, in its entirety. Something like the autobiography of Howard Hughes [1], where Hughes eventually said that he had no involvement, and did not know the writer. (This isn't actually the best example, because the book was pulled right before publication, but it's easy to imagine a scenario where it did get published. It came close.)

1: http://revisionisthistory.com/episodes/50-oh-howard-you-idio...

"This is a book full of things that I believe to be true. It therefore belongs in non-fiction" - random nutjob with a blog

In real libraries, they only stocked published books, and publishers decided what to publish. So the publishers acted as gatekeepers and fact-checkers (sometimes not so well, as the Howard Hughes example shows), and the library could accept their classification.

Now, anyone can write a book and add it to our hypothetical library. If the librarians aren't going to classify the books, then who does? Because someone has got to...