Does zakat apply to a rented apartment, knowing that the tenant refuses to vacate unless paid 40% of its market value by the owner?

You are not required to pay zakat on the rented apartment because it is not for trade. Zakat is only due on the rent if it reaches the nisab (minimum threshold) and a full hawl (lunar year) passes over it. The oppressive tenant must be reminded of the gravity of consuming people's wealth unjustly and the punishments it incurs. You are not obligated, in Islamic law, to give him money in exchange for vacating the apartment.