Is it permissible, Islamically, for a landlord to request a tenant to vacate an apartment when the landlord needs it, even if the lease term is not specified, and what is the ruling on the amount of money the tenant requests in exchange for vacating?

If the duration of the lease is not specified, then the lease is invalid (fasidah), which entails the annulment and voidance of the contract. The owner is entitled to the fair market rent for the past period, and the tenant is obligated to vacate the leased property. If he refuses, he would be usurping your apartment.

If the contract between you is on a monthly or yearly basis (an agreement to lease the apartment for a certain amount per month, or per year), then a group of scholars has deemed this method valid, and this is the prevailing fatwa (definition) with us. The contract automatically renews at the beginning of each month unless one of the parties refuses. If he pays his monthly installment, the contract becomes binding until the end of the month. If the month ends and you do not wish to renew the term, the lease terminates, and he must vacate the apartment.

His condition of a sum of money in exchange for vacating the apartment — in the case where the lease is invalid (fasidah) — is not permissible for him to stipulate or to take; because it is consuming people's wealth unjustly, for he is a usurper. However, if he refuses to leave except for the sum, and there is no way to remove him except by paying it, then it is permissible for you to pay it, and it is forbidden for him to take it, unless your soul willingly gives it.