Is a student considered sinful if they study their teacher for more hours than permitted, and is the knowledge gained during these extra hours considered unlawful, if Allah's saying, "Ask not about things which, if made plain to you..." does not apply to this situation?

If the teacher exceeds the time allotted by the administration, there is no sin upon you as a student, and it does not affect your knowledge. If this results in harm, students can file a complaint with the administration after alerting the teacher. However, if there is no harm and the teacher does this out of eagerness to complete the curriculum, there is no objection to it, and remaining for the additional time should be by choice.