Is it permissible to become a partner in a restaurant that sells pork and alcohol, knowing that the partnership does not require a monetary investment?

It is not permissible for you to participate with your husband's brother in his shop where he sells alcohol and pork, even if you do not contribute money, because this entails participation in sin and unlawful earnings, regardless of whether he sells them to Muslims or non-Muslims. As for shunning him and ceasing to speak with him, it is prescribed if it is effective in deterring him, and your husband must assist you. However, if it is not effective, it should be replaced with advice and reminders, while observing the Sharia-compliant guidelines for interaction. Your husband must also advise him.