Other mistake: At the end of the film when they arrive in NYC looking for the library, we know that they survived because they are inside with a roaring fire burning. But where is the smoke up above, as Jack Hall approaches? There must be a chimney, or else they would have all died of asphyxiation.
Question: Why would Sam and his friends go to the library?
Answer: Possibly because it was the closest building with height to it as they are about to be hit by a gigantic wave of water. There was no snow yet, so I don't believe burning books or snow was on anybody's mind yet. It turned out to be a great idea as snow soon starts to fall and those books were literally a life saver.
Answer: It was the closest building they could access. While the smarter move would have been to just go back to JD's apartment (which Brian and Laura suggest) it may have been too far a walk to get out of the flooding streets.
Why did they burn the books and not the wooden shelves that the books were on?
As for burning books rather than shelves, it was just easier. They would have had to expend more energy to break down the shelves into manageable size.
Factual error: When some people are burning books in the library, one man is adamantly protecting a Gutenberg Bible. Considering how rare and valuable Gutenberg Bibles are, it would not have been on the shelf or even readily accessible. Likely it would be locked in a safe or at least a showcase. Considering the shortage of time to collect fuel and the plethora of books available, I doubt they would break into a locked container to get another bit of fuel.
Suggested correction: According to Wikipedia, and other online sources, the Gutenberg Bible of the New York Public Library is publicly on display. It's shown in as being in a glass display case, there for anyone to see. There's no evidence that the book was taken for fuel. The first we see of it, it's being held by the person safeguarding it. It's possible they took it specifically to keep it safe. https://en.m.wikipedia.org/wiki/Gutenberg_Bible.
Suggested correction: He was the head librarian, and as such would have access to the safe/showcase. And he would want to keep the Gutenberg as close as possible to protect it.
That man clutching the Gutenberg was NOT a librarian, much less a "head librarian." He was just one of the civilians who came into the library to escape the climate onslaught. He wouldn't have access to something as precious and priceless as the Gutenberg. Original flaw remains valid.
Suggested correction: As already corrected, just because we can't see an outlet for the smoke doesn't mean there isn't one. White smoke against white snow is very hard to see.
Nik Rolls
But they were still burning the books, which meant the smoke should be black, if they stop burning the books, the smoke would have been white, which isn't the case in the movie.
Just looked at a boatload of videos of burning books, including one in a fireplace, and the smoke was indeed white.