What does the Bible teach about interracial marriages?

Here's the short answer: nothing in the New Testament prohibits interracial marriage. Christians are at liberty to marry whomever they wish--as long as the other person is also a Christian (1 Corinthians 7:39).

The Old Testament contained some restrictions on whom the Israelites could marry. These were meant to protect the people of God from pagan influences (Deut. 7:3-4) and to guard the purity of the Aaronic priesthood (Lev. 21:13-14). There were even some strict regulations about inter-tribal marriage, designed to keep the various tribes' inheritance from being lost by intermarriage (Num. 36:3-9). However, all such restrictions are explicitly annulled by the New Testament (Gal. 3:27-29).