Is it right for a Christian to marry someone who is not a Christian? The Bible answer is "no." Paul said in I Corinthians 9:5, "Have we not power to lead about a sister, a wife, as well as other apostles, and as the brethren of the Lord, and Cephas?" The fact that Paul specifies he had the right to take a sister (a Christian) for a wife, shows that it would not have been right for him to take a woman who was not a sister. Just like when God specified “gopher wood” in building the ark, that ruled out Noah using another type of wood.
Someone might reply that indeed, Paul had the right to marry a sister, but he didn’t have to, the same way Paul had a right "to forbear working" (verse 6), yet he chose to work (verses 15a and18). Their point is that a right doesn’t always imply a requirement, which is true. But the parallel in verse 5 is that Paul had the right to get married, but he could forgo his right to marriage if he so chose. However, if he did decide to get married, he could only have chosen a "wife that is a believer" (ASV); that is explicitly specified. Why else would Paul have stipulated a sister/believer?
Consider this illustration: Alabama state law says a 15 year old has the right to drive a car "with a learner’s permit and with a licensed adult in the car." We all recognize that is saying he has the right to drive a car, but since it is a right (not a requirement), he does not have to drive. However, the two stipulations specified are not optional. If the young man chooses to drive, he must have his learner’s permit, and he must have an adult with him at all times. Similarly, everybody ought to be able to see the parallel that though we don’t have to marry, the stipulation of "a sister" in I Corinthians 9:5 makes it a sin (a “transgression of the law” – I John 3:4) to marry a wife that is not a sister.
Since God is sovereign, if he wants to he can make it wrong for a Christian to marry an unbeliever, but if they do marry an unbeliever, wrong for the couple to divorce/separate (I Corinthians 7:12-13). In this case, I Corinthians 9:15 proves God chose to do exactly that. It is only left up to us to comply (Hebrews 5:9).
(while studying this question, please also consider the following other passages: I Corinthians 7:39, 15:33, Nehemiah 10:30, and Deuteronomy 7:3-4)