The only restriction I see in Scripture regards a Christian marrying an unbeliever. Marriage is an divine institution and ordained of God for all people. It is better for people to marry than to live together in an unmarried state for many reasons, but one of the chief reasons is that marriage does continue to promote the sanctity of marriage and the family.
I think when pastors are asked to marry unbelievers they should point out that they are not in the business of conducting marriage ceremonies, but that you would be glad to marry them if they are willing to come for a series of counseling sessions. Explain that you are in the ministry of building and promoting solid family relationships, etc. Other than the sentimental value or to please parents, getting married in a church will add no special benefits to their relationship or place God’s blessing on their marriage. If they are willing to come for counseling, then this give you an opportunity to share the gospel in the process of discussing the origin and purpose of marriage, the need of God, and so on.
What if they do not come to believe in the Savior, or what if only one of them does? If both do not, then the above still applies regarding marriage as a divine institution, but insist on sharing the truth at least about marriage as God designed it. If only one person comes to Christ, then that’s a different story and I think you would need to say, “I can’t marry you because…”
When dealing with unbelievers, the issue is not their moral behavior now (if they are living together for instance), but whether they will accept Christ. As you know, people aren’t saved and given new life by virtue of changed moral patterns. This is a matter of higher priorities taking precedence over lower ones.