As someone who has had to wade into an itellectual property fight before:
Copyright is only part of the equation. In your case, if you're writing fan fiction for a movie, you are just as likely to engage in trademark infringement. Whereas copyright is essentially protection from plagiarism, trademarks protect reputation. I don't know which movie you're writing about, but there is a good chance the title of the movie (and/or the name of the franchise) has been registered as a trademark, and so any work that invokes that name and describes a comparable plotline would be in violation.
And to be clear, copyrights and trademarks don't need to be registered to be protected. If someone wrote or created something, and they can prove it, then all of the IP rights are theirs.
But will you get sued for posting a story on this site? Only if the trademark owner sees you as a threat or a nuisance, probably.
A few years ago, there was a proliferation of Star Trek fanfiction, up to an including two independent recreations of the 1960s sets, and dozens of web episodes posted online for free. This continued for a while until one production, which was to be called Star Trek: Axanar, held a very successful crowdfunding campaign. This caught the attention of the trademark owner, CBS, which cracked down not just on Axanar, but everyone.
That being said, I doubt anything on CYS is likely to bring down such corporate wrath. There is already fanfic here on Star Wars, Saw, Super Mario, and god knows what else, and if those haven't gotten the site shut down, chances are no one is paying attention. Just don't make your fanfic too successful, I guess.
I should also point out that the phrase Choose Your Own Adventure is itself a registered trademark, currently owned by Chooseco LLC, based in Vermont. They recently sued Netflix for using the CYOA name without permission in a movie last year.