I know there are a lot of non-biblical arguments for the separation of church and state. Some Christians also support it on biblical grounds, but I've never looked into the matter myself. Which passages of the Bible support the idea?
Some people use Mark 12:17 towards this goal.
17 And Jesus answering said unto them, "Render to Caesar the things that are Caesar's, and to God the things that are God's." And they marvelled at him.
The argument goes something like this: The government is here to govern the society here on Earth. God has been given to us to govern our hearts, minds, and souls. So, we should allow the government to do what it does best and allow God to be in control of all the other things.
The argument is generally followed with the idea that we need to elect Christians into government and pray for our government officials, but that religion should be separate from the state.
Personally, I think this is a bit of a stretch. The more solid arguments are the non-biblical ones (prevention of the majority overriding the minority, insuring freedom of religion, etc.)
I think this article should help:
The Bible does not articulate a full-blown doctrine of the separation of church and state. Yet, its seeds are clearly present. Jesus at least foreshadowed the concept when he said “[g]ive therefore to the emperor things that are the emperor’s, and to God the things that are God’s.” (Matthew 22:21) Jesus’ behavior was consistent with his words. He never took a coin from Caesar or sought the help of Herod in his ministry and mission.
In many places, the New Testament outlines the contours of the separate realms of the kingdom of God and the kingdom of Caesar. The church is given the tasks of spreading the gospel (Acts 1:8), teaching doctrine (Matthew 28:20), and discipling believers (Ephesians 4:11-13). The state is divinely ordained to resist evil (Romans 13:3) and keep order (I Peter 2:13-15). Although these realms sometimes overlap and do not necessarily clash, the New Testament bears witness to a two-kingdom world — each with separate duties and each engendering different loyalties.
1 Corinthians 5:9-13 seems to tell Christians not to use man-made laws to force non-believers to live or behave according to Christian beliefs. Instead, it tells them to disassociate themselves with those WITHIN THE CHURCH who behave immorally, and to leave those outside the church to God's judgment.
1 Corinthians 5:9-13 King James Version (KJV)
9 I wrote unto you in an epistle not to company with fornicators:
10 Yet not altogether with the fornicators of this world, or with the covetous, or extortioners, or with idolaters; for then must ye needs go out of the world.
11 But now I have written unto you not to keep company, if any man that is called a brother be a fornicator, or covetous, or an idolator, or a railer, or a drunkard, or an extortioner; with such an one no not to eat.
12 For what have I to do to judge them also that are without? do not ye judge them that are within?
13 But them that are without God judgeth. Therefore put away from among yourselves that wicked person.
This is, to my knowledge, the Bible's strongest statement relevant to the idea of church and state separation.
Matthew 6:24 says that no man shall serve two masters for he will hate one and love the other or hold to one and despise the other. For me this is what it means to separate church and state: we cannot be Christians and politicians at the same time unless we are truly moved in Holy Spirit.
There are too many Christians who say it is our "duty" to vote or else we have nothing to say about how our government functions. I disagree. I do not vote because it would consume me. I would get swallowed up by the world. I would begin to serve government instead of my heavenly Father and would love one and hate the other.
Instead, I pray that all the matters of state be in the hands of my heavenly Father and his will be done. I prayed this last election for God to put it in my heart to vote if that was what he wanted of me. I did not vote and I have heard the Lord my God speak many things to me. I think if an individual is moved in the spirit to vote, then it is God's will for them to do so. However, if you vote out of duty, you do a disservice to God by serving two masters and not Him alone. If you are caught up in who this man is and what he said, and this is how you vote, then how do you serve God? There has to be a separation unless your involvement is the movement of Holy Spirit. It is only in this way that you serve God and God alone.