Once upon a time, people lost their heads to an executioner with a terrifying axe. (Check out Google images for "executioner's axe.") So, "to face the axe" means basically "to die." For a school, that would mean to go out of business. For an employee, it would mean losing his or her job.

I think, it means someone is fired or lost their jobs. For the schools, I think it means the school has to be closed/shutdown.