In the Old Testament the Jews are forbidden from marriage with non-Jews. See, for example, Deuteronomy 7:3-4. However that was on the basis of religious rather than racial purity, as it was clearly stated that the reason was to prevent the Jews from "following other Gods". It is also notable that in the Book of Ruth, Ruth is a non-Jew married to a Jew, and the Bible contains no commentary saying that this was a bad thing, although she clearly also becomes a follower of God. She is the ancestor of King David.
In the New Testament there is no equivalent restriction, since Christians are intended to be of every race: "In Christ there is neither Jew nor Gentile" and "make disciples of all nations".
In the past some Christians have claimed a prohibition on interracial marriage, based on either the Old Testament prohibition or a false understanding of Acts 17:24-26: ( "God ... hath made of one blood all nations of men for to dwell on all the face of the earth, and hath determined the times before appointed, and the bounds of their habitation ...").
Recently there has been a strong movement away from such a prohibition. Even ultra-conservative Bob Jones university removed its prohibition on inter-racial dating in 2000.
This article gives more information on the history.
EDIT:I originally said that Bob Jones University continued its ban on inter-racial dating until 2005. In fact it stopped in 2000, and apologized for it in 2005.