Must a student leave university if a late fee is imposed, given potential difficulties in securing an alternative source of income and the fear of falling into the unlawful, especially if other housing options impose similar fees?

If the fine you are asking about is a late fee on a debt owed to the university, then there is no sin upon the student for studying. He must strive to avoid the delay that leads to him being obligated to pay this fine. The sin lies with whoever imposed it and took it, not with the one from whom it was unjustly taken. So, focus on your studies and dismiss these thoughts.