Question: "How would you explain passages in the Old Testament such as Deuteronomy 13 to a non-believer, where God commands that false prophets should be put to death? How is this different from the Koran, which condones the killing of infidels, those who don’t believe in the teachings of Islam?"
Answer: When we consider the death penalty in God’s Law, we have to look at it from a different perspective than when we evaluate the laws of men. When we look at the laws of the Old Testament, particularly those that involve capital punishment, there are at least three main things to keep in mind (this is background stuff, foundational to a specific response to the questions):
1. These laws are given by God, not by man, and He has the right to impose the death penalty for any sin that He chooses, since those were the terms He set down in the beginning (as Ezekiel says, "the soul that sins shall die"). So we are all guilty before Him and none of us deserves, on our own merits, another breath in this world. He is just and has the right to call for our death at any time. That is how we understand the commands to the Israelites to destroy all the men, women, and children in Canaan during the conquest under Joshua - they were deserving, and God called upon Israel to carry out His judgment upon them in that unique period of history.
2. Even though every sin deserves the death penalty from God, in the laws He gave to Israel the Lord reserves that punishment for the most dangerous and offensive sins: murder, blasphemy, false teaching, some sexual perversions, hardened rebellion, etc. I would even argue that false teaching about God deserves the most severe penalty, since it causes others to come into sin, spiritual confusion, and destruction. Jesus said, "It would be better for him if a millstone were hung around his neck and he were cast into the sea than that he should cause one of these little ones to sin." (Luke 17:2). Peter says of false teachers, "These men are springs without water and mists driven by a storm. Blackest darkness is reserved for them." (2 Peter 2:17).
3. However, these laws involving capital punishment were given to Israel as a theocracy (a government ruled directly by God through His appointed rulers). The purpose of Israel as an earthly theocracy was to be a shadow of the coming Kingdom of God, which is now represented by the Church. The parallel of a sinner being "cut off" (put to death) under Old Testament law in the New Covenant era would be the excommunication of an unrepentant sinner by the authorities in the church.
Basically, there are three types of Old Testament law, and it is important to recognize the distinctions between them if we are to understand their purpose:
a. Moral Law - summarized by the Ten Commandments; these apply to all men at all times, and are God's ongoing revelation of His standards for our thoughts, words, and deeds.
b. Ceremonial Law - these were the dietary laws, cleansing rituals, worship rituals, and sacrificial practices overseen by the Old Testament priesthood, all of which were types which foreshadowed the person and work of Jesus Christ in redeeming us at the cross. The purpose of these laws has been fulfilled and so they no longer apply.
c. Civil Law - these are the laws given specifically for the theocratic government in Israel. The Westminster Confession calls these "sundry judicial laws, which expired together with the State of that people; not obliging under any now, further than the general equity thereof may require." In other words, we wouldn't apply these laws directly to our situation today, but there are principles in them that reflect the moral law.
So, based on that overview of Biblical laws, we understand that the church is the successor to Old Testament Israel as the people of God, the visible representation of the Kingdom of God. But civil and ecclesiastical authorities are now separate, so the church is not given "the power of the sword," or the authority to use physical force as a punishment for or constraint upon evildoers. We are not to tolerate false teaching in the midst of the church, but we have no authority from God to impose any physical punishment on false teachers (or anyone else). Our only authority and power are spiritual, based in the proclamation of God's Word. Stonings or "jihads" are not part of the church's calling from God today.
The question this doesn't address is whether or not a hypothetical civil government that is submissive to God's authority and favorable towards the Church should ever apply the principles ("general equity" - WCF) of Old Testament civil laws for Israel in today's culture. In other words, if God poured out His Spirit and brought a major revival to America, to the point that our elected officials in the federal and state government had a desire to make the laws of our land conform to the will of God revealed in Scripture, would it be right to impose physical punishments, even capital punishment, for spiritual offenses (e.g., false teaching, blasphemy, etc.)? This has been debated for centuries and was a hot, relevant topic during the Protestant Reformation and the establishment of the American colonies, but no society is close to considering these kinds of things today. It doesn't appear to be God's intent to establish a theocracy today - His focus is upon the growth of His spiritual Kingdom through the spread of the Gospel and the Church within the many different types of civil governments around the world.