My father-in-law apostatized from Islam; he gave up prayer and fasting and he even raised doubts about the verses of the Quran. I am not sure how I should treat him. Should I be dutiful to him and treat him kindly? Am I permitted to eat from his food or to feed him from mine? Is my wife permitted to inherit from him if he dies before her?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.

In principle, the Sharee‘ah ruling of the apostate is the same as that of any other disbeliever. The apostate should be avoided and you should not  socialize with him as long as he remains in that state. However, if there is benefit in maintaining ties with him such as calling him again to Islam, then it is permissible to be courteous to him and socialize with him as a means of encouraging him to revert to Islam.

There is no objection to allowing him to eat your food or to eat his food, unless it is the meat of an animal that he himself slaughtered.

In case of his death, neither your wife nor any other Muslims are permitted to inherit from him. Likewise, he is not to inherit from his would-be heirs who die before him. This is because according to the Sharee‘ah, a Muslim does not inherit from a non-Muslim and vice versa.

Allaah Knows best.