A wife disobeys her husband when he asks her not to go to a certain place, and as a result of him refusing her permission, she doesn't talk to him for 7 days consecutively. What is Islam's position on this?

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 ‘alayhi wa sallam, is His slave and Messenger.

If the situation is as mentioned in the question  - that the wife does not talk to her husband, goes out without his permission and disobeys him - then she is a disobedient wife, so she does not deserve her husband to spend on her or provide her with accommodation unless she obeys him. Indeed we have issued many Fatwas about the ruling of a disobedient wife and the methods of treating her, so please refer to them; they are as follows: 84120, 85402 and 83324.

If the treatment that is mentioned in the above Fatawas is of no avail, then divorcing her is better than keeping her with him as when Laqeet Ibn Sabirah complained to the Prophet, sallallaahu ‘alayhi wa sallam , about his wife's bad language, so the Prophet, sallallaahu ‘alayhi wa sallam, told him: "Divorce her." [Abu Daawood]

For more information about the rights of the husband and wife, please refer to Fatwas 83157 and 88304.

Allaah Knows best.