Can a person who is married to someone who has a valid green card be deported if his papers have expired?

Yes. Given the cited circumstances, a foreign national who remains in the U.S. after his visa has expired is considered "unlawfully present" and is subjected to deportation, per the Immigration and Nationality Act (INA). INA § 212(a)(9)(B); 8 USC §1182(a)(9)(B).