Which denominations teach that all Christians must obey all the 10 Commandments literally? Including, but not limited to, teaching that Christians must observe the Sabbath as it was understood by the Jews, as beginning at sunset Friday, and finishing sunset Saturday.
The Seventh-day Adventist Church believes that all 10 of the commandments are still binding on Christians. This includes the seventh day Sabbath which starts on Friday sunset and ends on Saturday at sunset.
Judging by the interchange in the comment section of this question, I would hasten to add that the Seventh-day Adventist church does not believe that the 10 commandments are binding on Christians as a means of earning their salvation.
The following is a rather lengthy quotation from one of the leaders of the Seventh-day adventist church but it expresses the spirit of the official church sentiments and teachings regarding the view that Christian's are to have of the law:
The keeping of these commandments comprises the whole duty of man, and presents the conditions of eternal life. Now the question is, Will man comply with the requirements? Will he love God supremely and his neighbor as himself? There is no possible way for man to do this in his own strength.
The divine power of Christ must be added to the effort of humanity: "For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh; that the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit."
Repentance toward God for our failure to keep his law, is the first step in the Christian life, while faith toward our Lord Jesus Christ claims the merits of his blood for the remission of sins that are past, and makes us partakers of the divine nature. The carnal heart, that "is not subject to the law of God, neither indeed can be," is made spiritual, and exclaims with Christ, "I delight to do thy will, O my God; yea, thy law is within my heart."
There are many who say they believe in Christ; but do they? Have they the spiritual mind, the mind of Christ, that delights in the law of God? They claim to be the children of God, but they do not the works of God. We cannot afford to make any mistakes in this matter, for our eternal interests are at stake.
A correct faith will be made manifest in godly works, and will bring the whole life into harmony with the law of God. Faith and works must go hand in hand. Christ referred the lawyer to the law, and inquired, "What saith the law? how readest thou?" And he showed that those righteous statutes require our perfect obedience. When, through the goodness of God, our attention has been called to the demands of God's commandments, and light shines on us from his word, we are to believe and obey from the heart.
Many put their own interpretation upon the words of God; but we cannot depend upon them. We must know for ourselves "what saith the Scriptures." An infinite price has been paid for our redemption, and ought we not to bestir ourselves to search the chart and prove to our souls that we are in the highway cast up for the righteous, and walking in the path of humble obedience? We are warned to "make straight paths for our feet, lest that which is lame be turned out of the way." We are examples to others, and if we pursue a wrong course, and lead others away from the path of right, we shall be held accountable.
We can see the importance, then, of having true faith, for it is the motive power of the Christian's life and action; but feeling is not faith; emotion is not faith. We must bring our very work and thought and emotions to the test of the word, and true faith will be profoundly impressed by the voice of God, and will act accordingly. The Signs of the Times, November 24, 1887, par. 2-6. Ellen G. White.
I can write all day about this topic because it is one that Christianity seems to be so mistaken about. The verses quoted in response to @Peter deserve a thorough answer and that I am willing to give but it is beyond the scope of this question's answer.
A big problem here is misunderstanding between "taking the 10 Commandments seriously", and "must obey them all", literally. The simple answer to the original, unedited question is that all Christian denominations claim to take the 10 Commandments seriously, but only some of them take them to such an extent as to teach that members must be seen to observe the sabbath-day law in particular ways, as a yard-stick of whether they are likely to be keeping the rest.
Raising the question of whether any Christian's salvation depends on keeping them all is not in the scope of the question. Neither is it in the scope of the question to detour into literal sabbath-day-keeping needing to be on one particular day or another. Despite the OP mentioning this point, the edited question is simply asking for a list of denominations that teach Christians to literally obey all of the 10 Commandments.
Seventh Day Adventism is one such main denomination today (to answer the question). Also, the Church of Jesus Christ of Latter Day Saints, though they seem to place more dire importance on "Thou shalt not murder" than the others, as they have stated that the only way a murderer can have his sin atoned for is by the shedding of his (or her) own blood. Add to those two all denominations that teach moral perfection to be a possibility for Christians, and the goal to which they are devoted to attain. Not too many of them around, but a few.
Others that, in theological theory, maintain the necessity of keeping all the "moral laws" in the Old Testament - which are summed up in the 10 Commandments but which exclude the ceremonial and civil laws that were also given to Israel via Moses - are: Catholicism, Orthodoxy, Lutheranism, Anglicanism, and most Protestant denominations.
However, it is here that the waters become murky. As one illustration; although the Jehovah's Witnesses agree that the moral laws are requirements, they quite disregard keeping the sabbath-day sanctified (set apart to God) and as a day of rest. They do not see that as a moral requirement. And a lot of people from the broad spectrum listed above think similarly; today millions of people who claim to be Christians have little regard for one special day a week set aside to God and for their own rest. Further, many others also view not lying, not misrepresenting others, not stealing, not coveting, and not committing adultery as fairly indifferent matters (given how they like to be entertained by all of that, as a moral yard-stick). Although - in theological theory - they are told to keep God's moral law, all of it, many clergy and laity rest content if they can just be seen to be law-abiding citizens, whether or not they fiddle their taxes, commit adultery, lie or steal etc.
Really, the question could have been more easily, and swiftly answered by asking, "Which Christian denominations teach that adopting a legalistic view of the 10 Commandments is condemned in the New Testament?" This does not equate with saying their moral content is not important, so that Christians can become licentious - it shows that no human on earth (apart from Jesus) can keep the moral law of God, therefore that law will enslave and condemn them, whereas Christians have been liberated by Christ and are commanded not to get themselves under the yoke of legalism again. They are no longer condemned, but must diligently use their freedom in Christ to glorify God. There are not too many of them around, either.
In addition to the denominations mentioned in other answers, there are various related “Church of God” denominations that teach that the Ten Commandments are, and always have been, binding on all of humanity (including the 4th commandment regarding God's Sabbath day).
The Ten Commandments is a 76 page booklet published by the Living Church of God (available in various formats, including a mailed printed copy, free of charge, with never any request for money).
Here is an excerpt from it:
THE TEN COMMANDMENTS ABOLISHED?
In the previous chapters of this booklet, we have explored the positive application of the Ten Commandments to every aspect of our personal lives as living, active laws. But today, many professing ministers and Bible teachers are wrongly proclaiming that the Ten Commandments are “done away”—as they say—or that they have been replaced by the “new” commandments of Jesus.
First of all, let us notice one of the all-important purposes for Jesus Christ’s coming to this earth in the human flesh. Isaiah prophesied of Jesus, “He will exalt the law and make it honorable” (Isaiah 42:21). Here we find that Christ came not to abolish the law, but to “magnify” it (KJV).
To exalt, or to magnify, has just the opposite meaning of changing or abolishing something. It means to reveal in the most minute detail—to enlarge upon. Certainly the life and teachings of Jesus do just that with the Father’s law.
Jesus said, “Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill” (Matthew 5:17). Jesus did just what these words imply. Both in His life and teaching, he fulfilled the law. He magnified it by His perfect example. He filled it to the full, passing beyond the mere letter to observe even the minutest spiritual intent and purpose of the Father’s perfect law.
Those who knew Him as a teacher could never charge Him with having substituted the traditions of men for the commandments of God. He obeyed the Ten Commandments in word and in deed. He taught and lived them as the perfect way of life.
He said, “Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of heaven: but whoever does and teaches them [even the “least” commandments], he shall be called great in the kingdom of heaven” (Matthew 5:19).
Certainly, we should all aspire to be “great” in God’s Kingdom, for we should want to overcome as much as we can and have the opportunity to serve the most we can! Therefore, we should earnestly and fervently strive to do and teach even the “least” of God’s commandments. Do you think the Sabbath commandment is “least”? If so, you had better do and teach God’s Sabbath just as He commanded, following Christ’s perfect example in keeping holy the seventh day—not the “day of the sun”!
THE WAY TO ETERNAL LIFE
When a young man came to Him asking the way to eternal life, Jesus said, “But if you want to enter into life, keep the commandments” (Matthew 19:16–19).
The young man asked, “Which ones?”
Jesus proceeded to list several of the Ten Commandments. Jesus Christ knew the way to salvation! He said that way was obedience to the law of God the Father and surrender to His will.
Jesus declared, “Not everyone who says to me, ‘Lord, Lord,’ shall enter the kingdom of heaven, but he who does the will of my Father in heaven” (Matthew 7:21).
Far from abolishing the Ten Commandments, Jesus obeyed them (John 15:10). Christ was the “light” that God sent into the world to show men how to live. After His death and resurrection, Christ sent the Apostles out with this command: “Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all things that I have commanded you” (Matthew 28:19–20).
The Apostles had been there when Christ told the young man to “keep the commandments.” They had heard Him magnify the commandments of God in what is called the Sermon on the Mount (Matthew 5–7).
The Apostles had witnessed the obedience of Christ to the Ten Commandments, and knew that His was the perfect example. Therefore, when Jesus Christ sent them out to every nation with the order to teach them all things He had commanded them, there could be no possible doubt in their minds but that this included the Ten Commandments of God.
Obedience to the Ten Commandments, then, was the very basis of the teaching of Christ and of His original Apostles. But what about the “new” commandments of Jesus? Did they not alter or abolish the necessity for literally keeping the Ten Commandments that were revealed in the Old Testament?
WHAT THE “NEW” COMMANDMENTS ACTUALLY TEACH
Millions of professing Christians have been taught that all they need to do is “love Jesus” or have the “love of God.” What is that “love”? How does God Himself tell us how His love is to be expressed? At the very end of the Apostolic Age, decades after Jesus’ resurrection, God inspired the Apostle John (Jesus’ dearest friend among the Apostles) to tell us, “For this is the love of God, that we keep His commandments. And His commandments are not burdensome” (1 John 5:3).
You must understand before Jesus Christ was resurrected he and we were all still bound by the law of Moses Debt once Jesus Christ became high priest and changed the priesthood it also changed the law of Moses to the law of Faith
Now I say, That the heir, as long as he is a child, differeth nothing from a servant, though he be Lord of all; But is under tutors and governors until the time appointed of the father. Even so we, when we were children, were in bondage under the elements of the world: But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law, To redeem them that were under the law, that we might receive the adoption of sons. And because ye are sons, God hath sent forth the Spirit of his Son into your hearts, crying, Abba, Father. Wherefore thou art no more a servant, but a son; and if a son, then an heir of God through Christ. (Galatians 4:1-7 KJV)
For the priesthood being changed, there is made of necessity a change also of the law. (Hebrews 7:12 KJV)
Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith. (Romans 3:27 KJV)
And, behold, a certain lawyer stood up, and tempted him, saying, Master, what shall I do to inherit eternal life? He said unto him, What is written in the law? how readest thou? And he answering said, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy strength, and with all thy mind; and thy neighbour as thyself. And he said unto him, Thou hast answered right: this do, and thou shalt live. (Luke 10:25-28 KJV)