What does the Bible say about interracial marriage and interracial dating?

There is nothing in the Bible that says it is wrong to date or marry a person of a different race.

Bible passages
There are a few incidental mentions of race in the Bible (e.g., that Ethiopian's skin was different, Jeremiah 13:23), but there is nothing saying one race is superior to another. Moses was married to a Cushite (Ethiopian) woman (Numbers 12:1-16), and God was angry with Aaron and Miriam for criticizing that marriage.

The Hebrews were forbidden to marry Canaanites (Exodus 34:10-17; Deuteronomy 7:3-4). The reason was because the Canaanites worshipped idols, not because of race; the Hebrews and Canaanites were racially similar.

The apostle Paul encouraged Christians not to marry unbelievers (2 Corinthians 6:14), but here again, the reason was religious, not racial.

Jesus and His apostles taught that we must respect and show compassion for all people of God's creation without regard to artificial distinctions like race and nationality. In His Parable of the Good Samaritan (Luke 10:25-37), Jesus told of a Samaritan man who was kind to a Jewish man even thought they were enemies of different religions, nationalities and ethnic backgrounds. At the end, Jesus tells us, "Go and do likewise."

The Apostle Paul banned all discrimination based on race, social status, etc. among Christians:

You are all sons of God through faith in Christ Jesus, for all of you who were baptized into Christ have clothed yourselves with Christ. There is neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus. (NIV, Galatians 3:26-28)

Racism
A long history of racism, particularly in the U.S., has made some people believe there is something immoral about dating and marrying outside one's own race. In fact, interracial marriage was illegal in some states until 1967, when those laws were declared unconstitutional by the U.S. Supreme Court.

A number of attempts were made to use the Bible to justify those bans on interracial marriage. Vague assertions were made that God intended for the races to remain separate. Some verses (Exodus 34:10-16, 2 Corinthians 6:14, etc.) were quoted in part or otherwise out of context in an attempt to show that God opposed interracial marriage.

Church doctrine
All of the mainstream Christian churches fully approve of interracial dating and interracial marriage.