Is it permissible to isolate a mentally ill person who makes sounds and murmurs, wears unclean clothes, and whose 'awrah is sometimes exposed in the mosque, in a nearby place until the prayer is over, or should he be left as he is?

There is no harm in removing an insane person from the mosque to protect the houses of Allah from impurity and vandalism, especially if he disturbs the worshippers or has impure clothes. If the Prophet, peace and blessings be upon him, ordered the removal of someone from whom the smell of garlic was found, then an insane person is even more deserving of removal. Jurists have stipulated the prohibition of an insane person from entering the mosque to prevent his harm to the mosque and the worshippers.