What is the ruling on taking money in exchange for relinquishing a free-of-charge rented government apartment, knowing that the apartment is not the employee's property and the exchange is done with the administration's approval but without informing them of the financial compensation?

The institution owns the property and you have the right to usufruct (right to use) the apartment, not the right to its utility or benefit. Therefore, you are not entitled to exchange it for compensation. Your right is solely to reside in it. If you vacate the apartment, you must hand it over to the institution or to whomever they instruct you to hand it over to, without any compensation.