Do not resist an evil person; but whoever slaps you on your right cheek, turn the other to him also. (Matthew 5:39 NASB)
Do any denominations, historical or current, take this command literally? Which ones?
No denomination simply ignores this verse. No major Christian teachers say to ignore Jesus' teachings. But given the different hermeneutical approaches of different people, there will be different ideas about how to apply his teachings.
John Stott wrote a widely used commentary on the Sermon on the Mount which addresses Matthew 5:39. Stott is highly respected in evangelical circles, and enlists Dietrich Bonhoeffer (an influential Lutheran) and Charles Spurgeon (a 19th century Baptist often called the "prince of preachers") to help make his points. Here's some of what Stott says about Matthew 5:39:
Let it be said at once, albeit to our great discomfort, that there will be occasions when we cannot dodge this command but must obey it literally. It may seem fantastic that we should be expected to offer our left cheek to someone who has already struck our right, especially when we recall that 'the striking on the right cheek, the blow with the back of the hand, is still today in the East the insulting blow' and that Jesus probably had in mind not an ordinary insult but 'a quite specific insulting blow: the blow given to the disciples of Jesus as heretics'. Yet this is the standard which Jesus asks, and it is the standard which he himself fulfilled. ... Further, before we become too eager to evade the challenge of this teaching and behaviour as mere unpractical idealism, we need to remember that Jesus called his disciples to what Bonhoeffer termed a 'visible participation in his cross'. ... In Spurgeon's arresting phrase, we 'are to be as the anvil when bad men are the hammers'.
Yes, but an anvil is one thing, a doormat is another. Jesus' illustrations and personal example depict not the weakling who offers no resistance. He himself challenged the high priest when questioned by him in court. They depict rather the strong man whose control of himself and love for others are so powerful that he rejects absolutely every conceivable form of retaliation.
The only limit to a Christian's generosity will be a limit which love itself may impose. For example the apostle Paul once 'resisted' (same Greek word) the apostle Peter to his face. Peter's behaviour had been wrong, evil. He had withdrawn from fellowship with Gentile brothers and so contradicted the gospel. Did Paul give in to him and let him get away with it? No. He opposed him, publicly rebuking him and denouncing his action. And I think we must defend Paul's conduct as a true expression of love. For on the one hand there was no personal animosity towards Peter (he did not punch him or insult him or injure him), while on the other there was a strong love for the Gentile Christians Peter had affronted and for the gospel he had denied. (Gal. 2:11-14)
Similarly, Christ's illustrations are not to be taken as the charter for any unscrupulous tyrant, ruffian, beggar or thug. His purpose was to forbid revenge, not to encourage injustice, dishonesty or vice. ... He teaches not the irresponsibility which encourages evil but the forbearance which renounces revenge.
The command of Jesus not to resist evil should not properly be used to justify either temperamental weakness or moral compromise or political anarchy or even total pacifism.
In that list is probably some things you had in mind for what it means to take Jesus' command literally (or "seriously" in the original post). So are there major denominations or movements who believe we should apply Jesus' command as "total pacifism"? There sure are. Pacifism is practically a founding principle of the various Anabaptist denominations (Mennonites and Amish) and Quakers. The non-denominational movement of "Red Letter Christians" also espouse pacifism because of Jesus' teachings. Pacifism in general is also pretty common in Christianity.
Christian anarchism is a subset of Christian pacifism. Christian anarchists believe that since the state is a purveyor of violence (via the police and military) it is inherently evil. Therefore Christians are not even to hold office. The Amish even say that joining unions, because of their coercive power, is a violation of Jesus' command. Christian anarchism was pioneered by the 20th century Russian Christian novelist Leo Tolstoy in his books What I Believe and The Kingdom of God is Within You.
The path shown by Jesus is a difficult one that can only be trod by true martyrs. A "martyr," etymologically, is he who makes himself a witness to his faith. And it is the ultimate testimony to one’s faith to be ready to put it to practice even when one’s very life is threatened. But the life to be sacrificed, it should be noted, is not the enemy’s life, but the martyr’s own life — killing others is not a testimony of love, but of anger, fear, or hatred. For Tolstoy, therefore, a true martyr to Jesus’ message would neither punish nor resist (or at least not use violence to resist), but would strive to act from love, however hard, whatever the likelihood of being crucified. He would patiently learn to forgive and turn the other cheek, even at the risk of death. Such would be the only way to eventually win the hearts and minds of the other camp and open up the possibilities for reconciliation in the "war on terror."
Some movements, as Wikipedia notes, are "nonresistant" in some instances but more accurately subscribe to "nonviolent resistance" as a philosophy. Martin Luther King, Jr. is quite possibly the most famous advocate of nonviolent resistance, and he based that philosophy on the teachings of Jesus, as you can read in his book Strength to Love, which gets high accolades from Stott. His application of Jesus' principles during the civil rights movement is widely praised by Protestants, Catholics, and even non-Christians.
I don't believe any denominations advocate "temperamental weakness" (being a doormat) or "moral compromise" in response to this Scripture, though it's easy to see why individuals may understand Jesus' words that way on a superficial reading.
Do any denominations take “turn the other cheek” as a literal command to be followed?
There are several denominations within the Anabaptist tradition that observe this practice such as the Amish and Mennonites.
Some Quakers have a history of non-violence. The Seventh day Adventists also.
There can be differences between "non-violence", pacifism, and conscientious objection to war.
The First Reading of the Third Form: The Simple Entrance, PALM SUNDAY OF THE PASSION OF THE LORD is taken from Is 50:4-7. Verse 6 has:
Isaiah 50:6 (RSVCE) 6 I gave my back to the smiters, and my cheeks to those who pulled out the beard; I hid not my face from shame and spitting.
Therefore the LORD himself who teaches turn the other cheek, himself turned the other cheek. The LORD has led by example; his disciple can do no less.
The history of the Church is rich in stories of martyrs and saints who have gone to their eternal glory imitating their LORD. There is a legend about the martyrdom of St. Lawrence (d. 10 August, 258), who was one of the deacons of the Roman Church, and who was one of the victims of the persecution of Valerian in 258, and the legend goes like this:
The prefect was so angry he told Lawrence that he would indeed have his wish to die—but it would be by inches. He had a great gridiron prepared, with coals beneath it, and had Lawrence’s body placed on it. After the martyr had suffered the pain for a long time, the legend concludes, he made his famous cheerful remark, “It is well done. Turn me over!” - Source: St. Lawrence | Saint of the Day | AmericanCatholic.org.
It is important to note that turning the other cheek is compatible with a just defense. [cf. Jn 18:23].
23 Jesus answered him, “If I have spoken wrongly, bear witness to the wrong; but if I have spoken rightly, why do you strike me?”