Is it permissible, in Islamic law, for the father's heirs to reclaim an apartment that the father had relinquished for an amount that does not cover its real value, and what are the financial implications of that?

We understand from the question that the father's relinquishment of the apartment to another, whether for compensation or without, was prohibited by the responsible authority. Therefore, what the father did is impermissible, and the money he took does not belong to him because he exchanged for something he did not own and was not authorized to do so. Consequently, the money must be returned to the entity that owns the apartment, so they can take the difference between the real and symbolic rent. As for the apartment, the heirs may benefit from it if the responsible authority permits, otherwise not.