What is the ruling on living and sleeping with a brother who mocks religion and becomes more obstinate despite being advised with evidence?

Your duty towards your brother is to advise him and remind him of Allah and the gravity of his mockery and ridicule of religion, and that his action constitutes apostasy from Islam. If you advise him, his actions will not harm you, and it is permissible for you to live and eat with him out of necessity. You must abandon his falsehood and leave him when he engages in such talk. If you are in a country that applies Sharia (definition), you must report his matter to the judiciary.