At my school, both situations have happened.
Well, in my opinion, I think that private schools will be more of the type to force students to drop for a number of reasons but public schools shouldn't. Those who are at the top of the class and performing extremely well shouldn't be impacted by those performing too poorly, but those in the middle will likely get dragged down. But if a student, who is not the best at a subject, chooses to stay then well, they made that choice. I have never truly heard that a teacher has the right to do that for real though.