All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not permissible for the landlord to force you to pay more than your monthly rent because it is forbidden to charge extra money for the delay in payment as this was the same condition of the people of the pre-Islamic era of ignorance which was either to pay now or the loan will incur interest.
As regards the landlord asking you to leave the house, then he has no right to do so until the end of the contract period.
It should be noted that it is not permissible for you to pay him this surplus amount which he is asking you to pay unless he forces you to do so and you have no other alternative than paying it out of fear of being harmed (by him). That is because the prohibition of Riba (interest and/or usury) includes both the one who consumes the Riba and the one who gives it to him, as the Prophet said: "Allaah has cursed the one who consumes Riba (i.e. usury or interest), the one who gives it to others, the one who records it and the one who witnesses it (its transaction).” He said: "All of them are equal in sin." [Muslim]
Allaah Knows best.