Do Christians need to obey Old Testament laws?

It is important to keep in mind that the Old Testament was written to the nation of Israel, not to the church.  The study of the divisions of the time in the Bible, also known as dispensations, helps distinguish between different relationships that man had with God.  Adam, Noah, Abraham all had different commandments from God based than did Moses or you and I.  See the FAQ section about Bible prophecy for more information about dispensations.  Also, listen to the following sermon on dispensations.  
The section of Law, from Moses’ day to the time of Christ contained several classes of law – civil, moral and ceremonial.  The New Testament amplifies the importance of the moral law of the Old Testament while annulling the ceremonial and civil laws.
Civil Laws 
The civil laws were the basis of the government’s legal system and economy.  Land, for example, reverted back to its original owner every 50 years in the year of Jubilee.  Slavery, or bondservants, could sell themselves to pay for debts for up to seven years of service.  The tithe was a tax on the people to support the tribe of the Levites and the service of the temple.  As a nation, God asked Israel to conduct war on his behalf (Joshua 1:2).  No nation today can legitimately claim this right.
Ceremonial Laws 
The ceremonial laws precisely defined the process of worship done in the Tabernacle and then later in the Temple.    Christ is the fulfillment of the pictures and shadows that were contained in the ceremony of Israel.  In this sense He put an end to the Old Testament law the sacrifice of himself once and for all.  Speaking of obtaining salvation by keeping the law or by faith in Christ, Paul says in three places that the ceremonial law is now done away with.
(3)  For, being ignorant of the righteousness of God, and seeking to establish their own, they did not submit to God's righteousness.
(4)  For Christ is the end of the law for righteousness to everyone who believes.
(21)  Is the law then contrary to the promises of God? Certainly not! For if a law had been given that could give life, then righteousness would indeed be by the law.
(22)  But the Scripture imprisoned everything under sin, so that the promise by faith in Jesus Christ might be given to those who believe.
(23)  Now before faith came, we were held captive under the law, imprisoned until the coming faith would be revealed.
(24)  So then, the law was our guardian until Christ came, in order that we might be justified by faith.
(25)  But now that faith has come, we are no longer under a guardian,
(26)  for in Christ Jesus you are all sons of God, through faith.
(13)  But now in Christ Jesus you who once were far off have been brought near by the blood of Christ.
(14)  For he himself is our peace, who has made us both one and has broken down in his flesh the dividing wall of hostility
(15)  by abolishing the law of commandments expressed in ordinances, that he might create in himself one new man in place of the two, so making peace,
(16)  and might reconcile us both to God in one body through the cross, thereby killing the hostility.
The civil laws of the Old Testament also do not apply since there is no physical kingdom of God on the earth today.  There is no throne of David, no structure of tribes in Israel.  The laws of your country may have some root in the Old Testament, but they are no longer seen as a whole.  By the time that Jesus was born, the national laws of Israel had already been replaced by those of Rome.  In Matthew’s gospel, written primarily to the Jews, the following conversation shows us that Jesus respected the civil government by Rome.
(15)  Then the Pharisees went and plotted how to entangle him in his words.
(16)  And they sent their disciples to him, along with the Herodians, saying, "Teacher, we know that you are true and teach the way of God truthfully, and you do not care about anyone's opinion, for you are not swayed by appearances.
(17)  Tell us, then, what you think. Is it lawful to pay taxes to Caesar, or not?"
(18)  But Jesus, aware of their malice, said, "Why put me to the test, you hypocrites?
(19)  Show me the coin for the tax." And they brought him a denarius.
(20)  And Jesus said to them, "Whose likeness and inscription is this?"
(21)  They said, "Caesar's." Then he said to them, "Therefore render to Caesar the things that are Caesar's, and to God the things that are God's."
(22)  When they heard it, they marveled. And they left him and went away.
Later, Peter wrote that all authority, whether Jewish or Roman was to be respected, even in respect to the civil laws of slaves.
(12)  Keep your conduct among the Gentiles honorable, so that when they speak against you as evildoers, they may see your good deeds and glorify God on the day of visitation.
(13)  Be subject for the Lord's sake to every human institution, whether it be to the emperor as supreme,
(14)  or to governors as sent by him to punish those who do evil and to praise those who do good.
(15)  For this is the will of God, that by doing good you should put to silence the ignorance of foolish people.
(16)  Live as people who are free, not using your freedom as a cover-up for evil, but living as servants of God.
(17)  Honor everyone. Love the brotherhood. Fear God. Honor the emperor.
(18)  Servants, be subject to your masters with all respect, not only to the good and gentle but also to the unjust.
Paul says many of the same ideas in Romans.  The governing authorities in Paul’s day were no longer priests and Jewish kings, but Roman governors and emperors.
(1)  Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God.
(2)  Therefore whoever resists the authorities resists what God has appointed, and those who resist will incur judgment.
(3)  For rulers are not a terror to good conduct, but to bad. Would you have no fear of the one who is in authority? Then do what is good, and you will receive his approval,
(4)  for he is God's servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God's wrath on the wrongdoer.
(5)  Therefore one must be in subjection, not only to avoid God's wrath but also for the sake of conscience.
(6)  For because of this you also pay taxes, for the authorities are ministers of God, attending to this very thing.
(7)  Pay to all what is owed to them: taxes to whom taxes are owed, revenue to whom revenue is owed, respect to whom respect is owed, honor to whom honor is owed.
Moral Laws 
The moral law of the Old Testament is actually amplified in the New Testament.  One could technically fulfill the moral requirements of the Old Testament without actually feeling any love or enthusiasm toward the commandments.  In contrast the New Testament teaches us to fulfill the spirit of the law.  We are therefore under the law of Christ (Galatians 6:2), which is the law of love. 
(36)  "Teacher, which is the great commandment in the Law?"
(37)  And he said to him, "You shall love the Lord your God with all your heart and with all your soul and with all your mind.
(38)  This is the great and first commandment.
(39)  And a second is like it: You shall love your neighbor as yourself.
(40)  On these two commandments depend all the Law and the Prophets."
Once again in 1 John, the commands are defined by love.  The “commandments” that he is referring to are obviously not fulfilled by paying taxes, preparing meals according to kosher law or by avoiding mixtures of wool and linen in garments (Deuteronomy 22:11).
(2)  By this we know that we love the children of God, when we love God and obey his commandments.
(3)  For this is the love of God, that we keep his commandments. And his commandments are not burdensome.
Nine of the Ten Commandments are clearly repeated in the New Testament.  The commandment to honor the Sabbath, commandment four, is part of the civil or ceremonial law.  We still teach the Ten Commandments in Bible classes and congregations because they summarize the moral responsibilities that God wants from us.  
Although nobody can keep these commandments (Romans 3:23), they are still useful to teach us how sinful we are and bring us to Christ for salvation (Galatians 3:24).
Romans 3:20  For by works of the law no human being will be justified in his sight, since through the law comes knowledge of sin.
Romans 7:9 I was once alive apart from the law, but when the commandment came, sin came alive and I died.
Romans 7:13 …in order that sin might be shown to be sin, and through the commandment might become sinful beyond measure.
The Old Testament ceremonial law is nullified by the work of Christ in salvation.  The civil law no longer applies since the Kingdom of Israel no longer exists.  The moral laws of the 10 Commandments, however, teach us that sin can only be remedied by a savior since we fail to fulfill them.
Shad David Sluiter