A friend of mine asked me why I think there are biblical reasons against Christians marrying nonbelievers (which is not to say that it's wrong for Christians to remain married to nonbelievers upon becoming Christian, since Paul explicitly gives instructions on that situation; see below). It took much longer for me to explain than people often do in these situations. The usual response is to trot out II Corinthians 6:14-18, which is not about marriage at all but idolatry. My friend also gave some objections based on those men of God who did marry outside of Israel in the Old Testament, often leading to good consequences (including some of them being in the line of Jesus). So I had to do some background work on the biblical theology of marriage and intermarriage. Here's what I came up with.
First, I'd be careful about saying that something is ok merely because it happens in the Bible without being described as good or bad. Many of the actions of the leaders in the book of Judges were at best morally ambiguous, but often the narrative itself doesn't explain why. It assumes a background in the law for you to know that what they're doing is bad. The readers at the time would have known, and anyone who knows the books of Moses would also understand. Another example is the case of the patriarchs who had more than one wife. The book of Genesis details how disastrous this was for their families, though it never clearly states that it was wrong. That has to be derived from the first couple chapters when God gave Adam and Eve to each other to be one flesh, which Jesus comments on in Mark 10 a little more clearly. These are both cases of something being wrong but not being said or emphasized as being wrong, because the purpose of those events being recorded was at least partly for something else.
Second, the cases in question (Joseph, Moses) were before the law's command not to marry people from foreign nations (outside the covenant community of Israel), and it doesn't say that she didn't first come to follow Joseph's God (note: the patriarchs in Genesis do command their sons not to marry Canaanites in particular, but Joseph's wife was Egyptian, not Canaanite). We just don't know the details on some of these cases. Also, it's not clear he was even clearly following God in as strong a way as we would have expected if he had been given the law beforehand. He didn't have any of that, just the stories told by Jacob, Isaac, and Abraham about what God had promised and what he'd done. It's not clear that he could be held responsible for the same things later generations would, so it's not clear that we should expect every issue that would be important later to be as important in presenting his life. Since we know that all have sinned, we do know that he did sin, though. We just don't have any statements of what those sins were.
As for the good consequences argument, I'd be even more hesitant. There are many cases in scripture of God using something for good that was meant for evil. Joseph's brothers sold him into slavery, but Joseph says to his brothers that God used it (Genesis 50:15-20). Isaiah 10 calls the king of Assyria a tool in God's hand for judgment on Israel, and yet the guy is moral responsible for the evil he does against God's people. Judas and the Jewish leaders are a prime example of doing something evil that God uses for good. We can't let good consequences affect whether we judge an action to be good or bad. I do think God worked through intermarriage with Gentiles. Matthew's genealogy of Jesus shows three (I think) such marriages in Jesus' line (or rather Joseph's line, which we can't connect genetically with Jesus, but it is the royal line Jesus was adopted into). We do know that Rahab and Ruth were genuine converts who worshiped and served the God of Israel whole-heartedly. I believe Bathsheba also was. (I might as well mention Tamar and Bathsheba, both of whom were in the line of Jesus due to sinful actions, which underlines the point of the previous paragraph.)
The main argument I would start with comes from the principle behind intermarriage with the nations in the land who didn't follow Israel's God. Moses' wife, Rahab, Ruth, and Abigail were all Gentiles but presumably followed Israel's God. We assume Rahab and we know Ruth explicitly committed themselves to follow Israel's God and had genuine conversions. According to the basic principle for why intermarriage was prohibited, these would not be problems. Even if these weren't cases of genuine conversion, the two points above remain. Just because someone otherwise good did something, that doesn't mean it's ok, and just because something leads to good consequences doesn't mean God would condone it.
The main argument is based on the reason for intermarriage with other nations. Exodus 34:15-16 raises some serious concerns about taking a wife from the inhabitants of the land who would then worship their gods and lead their husbands also into worshiping other gods. In the New Testament it becomes clear that the temptation toward idolatry is not just the temptation to another religion but includes things like being influenced by the pattern of the world such that your priorities for something created become higher than your priority for the creator. I think it's easy to see on reflection that marriage to someone whose basic value system is not rooted in the things of God can lead to a shift in one's priorities over time away from the things of God. We see the effects of this in Solomon's life in I Kings 11:1-8 and in Jeroboam's in I Kings 16:31. Nehemiah 13:26 also mentions this about Solomon, noting that he was otherwise a great king.
The command is repeated in Deuteronomy 7:3-4. The Canaanites were also considered ritually clean from being outside the covenant, and marriage outside the covenant would make someone unclean (Leviticus 21:4). Uncleanness in the Torah is symbolic of a spiritual principle. It's symbolic of being outside God's grace working toward salvation in our lives. So the prohibition on intermarriage not only protects us from a higher degree of influence from the values of the world, the flesh, and the devil but it also indicates our desire to be united (Genesis 2 -- the two become one flesh) with someone who is spiritually clean as we have been made clean in Christ. I Corinthians 6:15ff. says that one who joins himself to a prostitute becomes one with her, and one who joins himself to the Lord is one spirite with him.
Therefore one who unites himself with a prostitute is uniting a part of Christ's body with a prostitute and thus symbolically uniting Christ with a prostitute. Prostitution language is used throughout the Bible for those who spiritually serve other gods (even in some of the passages above about intermarriage). Uniting oneself deliberately with a spiritual prostitute then unites Christ with a prostitute. This is the theological background behind II Corinthians 6:14-18, which doesn't deal primarily with marriage at all, but the same principles behind that passage also apply to marriage, as the rest of scripture makes clear.
In I Corinthians 7:12ff. Paul explains situations with one person who is a Christian married to someone who isn't, but these were generally cases when one person became a Christian after already being married. His instruction then was to remain married if for no other reason than for the sake of the other person's potential salvation (because leaving at that point would almost guarantee that the person would consider Christianity to be a bad thing, causing people to leave their spouse). Many other reasons are available for staying together, not least because of a commitment made previously and because they are already one flesh. The only exception is if the other person insists on leaving, and then in that case to proceed according to the law if reconciliation is impossible. There's considerable debate over what Paul means when he says (v.14) that the unbelieving husband or wife is sanctified through the believing husband or wife, but when he goes on to say that it's for the sake of the children, my suspicion is that it means the children are regarded as part of the covenant community even if one parent is not in it, since there's one parent committed to raising the child up in the ways of the Lord.