Is it permissible for him to stay in the same house with his divorced wife?

If a man has given his wife the last of three divorces (talaaq), or he has divorced her once or twice and her ‘iddah has ended, then she becomes a “stranger” to him (non-mahram) and it is not permissible for him to be alone with her or to touch her or look at her. 

Undoubtedly if they stay in the same house, it will be difficult for him to adhere to these shar’i guidelines such as not being alone with her and her having to observe hijab before him as with all other non-mahram men, unless the house is large and it is possible to section off a part of it for him, with separate facilities and a separate entrance for him. But if they are both in one house, with the same entrance and the same facilities, then it will be very difficult to avoid the haraam things mentioned above. 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: The woman who has been thrice divorced is a stranger to the man like all other non-mahram women, so the man cannot be alone with her, just as he cannot be alone with any other non-mahram woman, and he cannot look at her, just as he cannot look at any other non-mahram woman. End quote from al-Fataawa al-Kubra (3/349) 

The Standing Committee for Issuing Fatwas was asked: My father got sick and went into hospital. After he came out, he got sick again and had his lower leg amputated. Praise be to Allaah for the decree of Allaah. Then he became paralyzed and cannot sit up. He is with me in my house … he asked my mother who is married to him, he said: You are divorced. Then he said: Forgive me and I will forgive you. My mother is still in the house and she cleans him because he is unable to go to the toilet, and she feeds him, because we are all at school and at work. 

They replied: 

If this divorce was a third divorce, then it is not permissible for your mother to stay with him and uncover him and touch him, because she is a stranger (non-mahram) to him. But if the divorce mentioned was a first or second divorce, then your mother is regarded as recovably divorced, and he may take her back so long as her ‘iddah has not ended, and she has the same rights as any other wife. She may serve your father and touch him. But if her ‘iddah has ended and he did not take her back by saying words to that effect or having intercourse with her during the ‘iddah period, then she is a stranger to him, and it is not permissible for her to stay with him and for him to be alone with her, unless a new marriage contract is done. End quote. 

Fataawa al-Lajnah al-Daa’imah (20/226) 

And Allaah knows best.