It isn't illegal for the board of directors to keep him on. I would question the board's judgment but it would not be illegal! However, the President of the United States must be a natural born citizen by law. To keep a usurper on as POTUS would be illegal. It violates the highest law of the land. So, your analogy doesn't hold water.
The woman @ MIT was let go when her credentials or lack thereof were exposed, remember? She had been running the place.