What should he do with a tenant who does not want to leave the house?

Firstly: 

The reason why such problems arise between landlord and
tenant is the failure to adhere to the laws of Allaah when drawing up
signing the lease. One of the things in which many people go against
sharee’ah, causing them a lot of problems, is not defining the period of
rental, because defining the period of rental prevents disputes and protects
the rights of both sides; it obliges the landlord to let the tenant stay in
the property and it obliges the tenant to pay the rent to the landlord for
the duration of the rental period, and it obliges him to adhere to the terms
of the lease and pay the rent even if he does not use the property. 

Ibn Qudaamah said: 

There is no dispute among the scholars concerning the fact
that it is permissible to rent out property. Ibn al-Mundhir said: All the
scholars from whom we acquired knowledge are unanimously agreed that renting
out houses and animals is permissible. 

And it is only
permissible to rent them out for a specific, defined period, after viewing
and defining the property in question, because it cannot be known otherwise.
It is not permissible not to define the timescale or describe the property.
This is the view of al-Shaafa’i. 

Al-Mughni, 5/260. 

The correct view is that the rental period does not have to
be limited to a specific length of time, rather it can continue for months
or decades, so long as that is with the agreement of both parties. 

Ibn Qudaamah said: 

If the rental is to last
for a period it should be known, such as a month or a year. There is no
dispute concerning that as far as we know. Because the timescale is
important with regard to the property to be rented, so it must be known,
just as when selling a commodity that can be measured, it should be sold by
measure. 

Al-Mughni, 5/251. 

And he said: 

The maximum period of renting cannot be defined, rather the
property that is rented may continue to be rented so long as it is still
sound, even if the period is lengthy. This is the view of all the scholars…
because Allaah tells us that Shu’ayb (peace be upon him) said: 

“on condition that you
serve me for eight years; but if you complete ten years, it will be (a
favour) from you”

[al-Qasas 28:27]

The law of the nations
before us is our law too, unless there is evidence that it has been
abrogated.  

al-Mughni, 5/253 

Secondly: 

With regard to the death of either the tenant or the landlord
before the expiry of the rental period, the majority of scholars are of the
view that the contract is not annulled by the death of either party. The
Hanafis and some of the Taabi’een held a different view and said that the
contract is annulled, unless the heirs of the landlord agree to that, and
that they have the right to demand that the rental property be vacated. 

The correct view is the view of the majority, so the contract
remains binding on both parties for the duration of the rental period. 

In the book al-Ijaarah, al-Bukhaari is quoted as
having a chapter in his Saheeh entitled “Chapter: If a person rents
land and one of them dies. Ibn Seereen said: The family do not have the
right to make him leave until the period (of the rental agreement) is ended.
Al-Hakam, al-Hasan and Iyaas ibn Mu’aawiyah said: The rental continues until
the end of the period agreed upon. Ibn ‘Umar said: The Prophet (peace
and blessings of Allaah be upon him) gave Khaybar to its people in return
for half of the harvest. That was at the time of the Prophet (peace and
blessings of Allaah be upon him) and Abu Bakr, and the beginning of Umar’s
caliphate, and there is no mention that Abu Bakr and ‘Umar renewed the
rental contract after the Prophet (peace and blessings of Allaah be
upon him) died. End quote. 

Al-Haafiz ibn Hajar said: 

The words “Chapter: If a person rents land and one of them
dies” mean: is the rental annulled or not? The majority view is that it is
not annulled. The Kufis and al-Layth were of the view that it is annulled…
they agreed that the rental is not annulled if the person in charge of the
waqf dies, and this is the case here.  

The words “The family do not have the right to…” refer to the
family of the deceased. 

“to make him leave” means, to make the tenant leave. 

The point here is to prove that the rental agreement is not
annulled if one of the two parties dies. This is the apparent meaning and is
referred to by the words “and there is no mention that Abu Bakr and ‘Umar
renewed the rental contract after the Prophet (peace and blessings of
Allaah be upon him) died”. 

Fath al-Baari, 4/463. 

Thirdly: 

With regard to what you
should do about the tenants who refuse to leave, we advise the following: 

1 – You should explain to them that the contract between you
and them is not acceptable according to sharee’ah because it does not
mention the time period for the contract. We have stated above that the time
period must be stated, and there is no dispute among the scholars on this
point. 

2 – You should give them sufficient opportunity to vacate the
property, and this period should be stated clearly to correct the previous
mistake and there should be a new contract with a set time period. 

3 – If they refuse to leave, then you should tell them that
they are usurping the property and are sinning, and you should tell them of
the punishment for wrongdoing and the ruling on those who usurp property. 

4 – You should bring in some influential wise men who could
speak to them, or some of their relatives or acquaintances whom they will
listen to. You can agree to pay them some money to leave the apartment. It
is haraam for them to consume this money, but as far as you are concerned,
if the person who is wronged cannot get his rights except by giving money to
the wrongdoer, it is permissible for him to do that. 

See question no. 40272. 

5 – If that does not work either, then you have three
choices: You can pray against them – for the prayer of the oppressed is
answered; or you can let them off; or you can leave their case to Allaah to
take your rights from them for you. 

And Allaah knows best.