A not frum person is married to a non-Jewish woman in chutz l’aretz. He wants to put a mezuzah on the door of their house. Seemingly this would be considered a “shutfus” and would be patur. However does he “lose” anything by puttng one up. Or is it ossur to put a mezuzah on his door in such a case when he lives with a non-Jewish woman?
The halachah of putting up a mezuzah for a house of shared Jewish/non-Jewish ownership (or occupancy) is disputed by rishonim. According to the Mordechai, there is no obligation, whereas the Rashba writes that there is an obligation.
The Rema (286:1) rules according to the Mordechai, and therefore there is no obligation to put up a mezuzah. However, it is permitted to put one up, and this should be done without a berachah.
The fact that the person is living with a non-Jewish woman does not mean that a mezuzah should not be put up. Living with a non-Jewish woman is of course not permitted, but it does not prevent the person from fulfilling other mitzvos.