The key verse emphasizing that believers should only marry fellow believers is I Corinthians 7:39. Written in a context of questions about marriage, it says that one should marry only “in the Lord”, which limits prospective marriage partners to those who are fellow believers.
So if a believer desires to marry an unbeliever, the elders of his or her church need to admonish them against doing so. The pastors in the church should not perform the wedding. If the person persists in that choice, the wedding should be limited to a civil ceremony.
The elders will need to decide whether they should start the process of disfellowshipping such a person. If they choose not to do so, they should be very strict with such a person, not allowing them to take any role—not just a leadership role, but any kind of role, even a servant role—in the congregation.
The church should never convey the idea that the action of a believer choosing to marry an unbeliever can be accepted in any form in the local church.
I realize it is hard for many churches to be so strict these days. However, the believer answers to a higher authority than the secular civil standard. Disobeying the biblical standard should not be treated lightly.