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.
For the Friday Khutbah to be valid, it is a condition that the speaker be Muslim. Since it is a communal obligation, the Friday Khutbah is invalid when it is given by either a genuine disbeliever or an apostate. The scholars of Fiqh stated some conditions for the Khutbah to be valid. While some of these conditions are agreed upon, others are controversial. When any scholar's conditions are applied to a non-Muslim, it is inevitable that at least some of them are not fulfilled, and this renders the Khutbah invalid accordingly. Many scholars did not mention the condition of Islam, as it goes without saying and it is known to all, while other scholars mentioned it for further clarification. However, none of the scholars of Islam said that the Friday Khutbah is valid when it is given by a non-Muslim. A person who makes this claim must provide evidence, and he will never be able do so.
Ibn Al-Wardi is one of the prominent scholars who mentioned the condition of Islam. Mentioning the Jumu‘ah's conditions, he said: "The congregational (Jumu‘ah) prayer needs forty believers to be valid; meaning for the prayer and the Khutbah." Shaykh Zakariyya Al-Ansari objected to his mention of this condition arguing that it is conditioned for all prayers. But the prominent scholar Ibn Al-Qasim responded: "The words of the author (Ibn Al-Wardi) signify that this condition also applies to the Khutbah; thus, it is required for it as well."
Consequently, it is conceded by all Fiqh scholars that the one giving the Friday Khutbah must be Muslim. It is improper and unacceptable that some people, with neither knowledge nor understanding of the religion, come at the end of times to speak about the applicability of Sharee'ah rulings. Meanwhile, none of them dares to discuss other subjects, medical issues and the like, unless he spent his entire life studying it. This is due to the continuous lack of respect towards the Sharee'ah; until it became violable by all. There is neither power nor strength save by Allaah Almighty.
As for the issue of the Imaam, if he commits this disobedience in secret, then you are not permitted to disclose this to the people. However, you are indeed permitted to do so if he sins openly. According to the preponderant opinion of most scholars, the prayer behind a disobedient person is valid.
Allaah Knows best.