This is a good question. The answer is simple, God never changes. This means every word of the Old Testament is still applicable to us today as Christians as it was to the Jews to whom it was first given. In the New Testament, God does not change His Word, rather, He reveals the purpose of the Word in regards to man. As such, we must look at this to understand it.
Paul explains this entirely in the book of Romans. I suggest you read through this book so that you may understand this topic better. For now, I will give you a summarization that hopefully help you understand.
To understand the purpose of the law, we must rewind back to the beginning. Why did God create humanity? How did we fall from that purpose? Why did God give us the law? And what did Christ do that changed this?
PURPOSE OF HUMANITY
God created humanity for His glory; that He would rule over us and by doing so, we would reflect Him. Simply said: God created humanity for Him. We were created for His pleasure and joy so that He may rule over us as a loving father, and by doing so, we would be a reflection of Him.
In the garden, however, we fell from the glory of God. When Adam and Eve disobeyed God, they did not reflect God, they reflected something else. As such, they sinned and became sinful. Before they ate from the tree, God was their father and He ruled over them in love. After they ate from the tree, they left the rule of God, and entered the rule of Satan who controlled them through deception.
This is important to realize. We left the kingdom of God. We were no longer under His rule and authority. Because of this, we fell. This means that our nature and being changed. We were no longer a glorious being. We were a defiled creature as we could not produce any light or glory of our own. This means that not only did Adam and Eve sin, they became sinful. Their nature and being was sinful.
Just in case you don't yet understand this, I am trying to show that sin was not just an act. It was like a disease that plagued our soul. We are sinners, not just because we do bad things, but because our whole being is filled with sin. A dog barks because it is a dog. A sinner sins because it is a sinner. A dog cannot change itself. A sinner cannot either.
THE PURPOSE OF THE LAW
Let us fast forward to the law. God's purpose for us was to be a reflection of Him. We were created for God's glory to be a reflection of Him in all ways. This means we are to be like God. We are to Holy, just as God is Holy. We are to be good, just as God is good. We are to be just like God.
However, as shown above humanity is sinful. We are sinners. We cannot change ourselves. We cannot be like God. The problem becomes, since we have fallen short of the glory of God and do not know God, how can we know that we exist in this state? How could we ever possibly know that we are sinful?
As such, God gave us the law to show us how sinful we are. The purpose of the law was not to change us, but to make obvious this problem of sin. When we try to obey the law and find that we fail, the hope is that we will realize our condition. We are sinful. We cannot obey the law. We cannot be like God. When we realize this, we seek salvation through Jesus Christ.
The law was given to show our sin. The law was not given to rule us. Indeed, the law was original given as a set of commandments. They were to be obeyed. Everytime we disobeyed the law, we knew that we had sinned. However, the law was not given to rule us. This means, the law was not supposed to be our God. We were created to live under God's rule. We were created for God, not for the law.
However, since we were far from God and did not know Him, God gave us the law to show us what we should be like. If any person did not have the disease of sin, if they were born without sin and sinless, then they would obey the law naturally without trying. One person in our entire history did this. God Himself was born as a man named Jesus Christ. He was born without sin. This means His very being was sinless. Jesus did not have to try to obey the law. The law was His nature.
To illustrate this point even further, let me give you an example. Let's say in our society, nobody ever drove too fast. Everyone naturally drove slowly. If everyone drove slowly, would we need to create speeding laws? No. Obviously not. We create speeding laws because people speed. The law is given to restrain people from following their natural tendency to speed. The law restrains through reward and punishment. People will slow down whenever they see a police officer because they fear punishment. However, when they think they won't get caught, they speed. So, by this law, we see there is a naturally tendency within humanity toward speeding. However, if a person was born with a natural tendency to drive slowly, would that person ever be fined for speeding? No. Because that person's nature is good.
The very nature of Jesus was good because He is God. Jesus fullfilled the law because He didn't need the law. He was a perfect and complete being. He never broke a law because His being was good.
The law was given to us sinful people because we needed the law to show our sin. Without the law, we could go around being sinful and never know. However, the law makes us accountable. When the law is applied, it shows our sin and makes us accountable for our sin. In this, the law also shows the justice of God because when He judges everyone, He will have the law to show His judgment is right. The law was given, and obviously, it was not obeyed. And so, this shows God's justice in giving righteous punishment.
Here are some scriptures relating to the purpose of the law in showing humanity our sinful nature: Romans 3:19,20; Romans 7:7; Galatians 3:24
HOW WE ARE CHANGED BY CHRIST
Obviously, the law could not change a person. Our problem was not an external one, but an internal one. We break the law of God because we are sinful. Our being needs to be changed. We need to be born again. God knew of our dilemma and through Christ He offered a solution to our problem.
Jesus was born a sinless man. He was also born an Israelite to whom the law of Moses was given. This means that Jesus was born under the law. Being that Jesus was sinless, He did not break a single law. He obeyed every law, and by doing so, He fullfilled the law. (Matt 5:17)
The law said (Leviticus 18:5) that if we keep the decrees and laws, we will live by them meaning that we will have life. However, the problem is that if we break one law, we have broken them all (James 2:10). Because we are sinful, we have broken all the laws, and so, this caused death to work in us. The law was supposed to bring us life, but instead, it gave death even more power.
Jesus did not break any law, and so, in Him was life. Death was alive and at work in us, but life was at work within Christ. So, when Christ died on the cross, He paid the ultimate sacrifice. He gave His eternal life to pay our eternal debt. As a result, the ransom was made.
However, something else important happened with Christ on the cross. When a person dies, they are no longer under the law. The law, as Paul points out, is only applicable while a person lives. (Romans 7:1) When Christ died on the cross, He was no longer under the law which produces sin and death. This means that anyone who is baptized into the death of Christ on the cross, whoever shares in His death, is also set free from the law. They do not live under the law any longer.
As followers of Christ, this is important. This means that we do not live by the law. We live for Christ. We are not controlled by the law. We are controlled by the Spirit. This is awesome news. When the Son sets us free, we are free indeed.
However, this brings up another question, if we don't live by the law do we still have to obey the law?
The death of Christ on the cross sets us free from the law. The resurrection of Christ makes us alive to God.
When Adam and Eve sinned, they left the kingdom of God. They were no longer under the rule and authority of God as their Father. Later, God would give the Israelites the law. The law, however, was not given to rule us. It was given to show us that we were sinful and in need of salvation. As such, man was not made for the law, but law for man. (Mark 2:27) We were created to live under the rule of God, in the kingdom of God.
In order to enter the kingdom of God, we had to be set free from sin and death. Firstly, we were prisoners to death because of the debt of sin. This debt had to be paid. Christ paid this with His death. Secondly, we are prisoners to sin because our being is sinful. Therefore, we need to be born again. We need to experience a change of being. As such, this is fulfilled when we share in the resurrection of Christ. When we are born again, our sinful nature is changed. We are no longer sinful but instead live for God.
When we are born again, we are raised to new life. We live for God under His rule and authority. He becomes our heavenly Father. As such, when this happens, the law is fulfilled in us.
To explain this further, Jesus said that He did not come to abolish the law, but to fulfill it. The law was only a problem for sinful humans because we are sinful. However, when our sinful nature is removed, the law should be fulfilled naturally within us. We should live like Christ who obeyed the law naturally. We should not need to live in restraint. We should not need to fight with our inner being to obey the law. Instead, we should naturally do what God requires because the Holy Spirit is at work within us.
Before we go on, there is something you must understand in regards to the law. The law is external and it requires interpretation. God is greater than the law, and He established the law according to His will. This means that the law is representative of His person. When earthly humans approach the law, we must interpret the law trying to understand it. For instance, the laws says do not murder. But what does that mean? What is murder? Do we murder someone when we accidentally kill them? Or is murder intentional?
Since the law is external, it requires interpretation. In this interpretation, we should be seeking God Himself. We should say, "What did God mean by this? What is God's Will?" However, the sinful nature is against God. It does not seek after God. So, when you take the Holy law and give it to sinful people, what happens? They use it for their own advantage. They manipulate the law to suit their own sinful ways. This is epitomized by the Pharisees. They didn't care about God or His ways. They used the law to make themselves look good, and to maintain power over others. They used it for their advantage.
The Pharisees are an example of this, but truly, this is a problem with all humanity. We all abuse the law of God. We do this because we are sinful.
When a person is born again and transformed by God, they do not need to interpret the law any longer. As shown above, we should seek God's will in the law. When we are born again, we should continue to seek God's will, but it is made manifest in us by being controlled by the Holy Spirit. When we are controlled by the Holy Spirit, we naturally do the Will of God because the Holy Spirit controls us. This means that we may not mentally understand what we are doing or why, all we know is that God is in control. We have faith in Christ to be our Lord, who controls our actions by His Spirit.
This means that the Word of God no longer exists externally as a law or command. As Christians, the Word of God lives in us. (1 Peter 1:23; 1 John 2:14; John 15:4)
WHY DO CHRISTIAN'S STILL SIN?
Of course, being this is true, a question arises: Why do Christians still sin? If indeed we have been born again, and God is working in us, then Christians should naturally obey all the laws. They should never sin.
The apostle John also spoke about this very subject in the book of 1 John. Indeed, if Christ is within us, then we should no longer continue in sin. (1 John 3:9) In the eyes of God, we are sinless. However, you must understand that we have hope in Christ. We do not hope for something we have already received. (Romans 8:24)
When a Christian is born again, they receive a spiritual transformation. However, they still exist within this world with sinful flesh. As such, we are caught between two kingdoms. We are not of this world, but we live in this world.
A person who has been born again is not longer of this world. They have entered into the Kingdom of God and now live for God. However, this person also awaits a future day when our whole being, our flesh and all, will be resurrected or transformed into glory. As Christians, we hope for this day. Until then, we are caught between two Kingdoms. Spiritually, we live for God and in the Kingdom of God. Physically, we live in this sinful world whose prince is the Devil. As such, we live in a war zone.
Christians belong to Christ. He is their Lord and King. However, we live in a world whose prince is the Devil. And so, we constantly battle against the Devil and the dark powers of this world.
Because of this, Christians often make mistakes. Our goal as Christians is to bow down and offer ourselves entirely to Christ. When we do this, the Holy Spirit is able to control us. Then we produce good fruit. However, the Devil seeks to keep us from producing good fruit. Therefore, we are sometimes lead astray. We are sheep whose shepherd rules only in love. Though Christ is our Lord, He will not rule in force. He waits for us to willingly lay down our life for Him. He will not oppress us or force Himself upon us. He stands at the door and knocks. (Revelation 3:20) It is left to us to let Him in.
So, though a born again person has a renewed spirit, they may still fall into sin. However, it is not their nature to sin. They have merely erred, and Christ, their Lord, will set them straight through loving discipline. (Hebrews 12:4-17)
Based upon this, we go back to your question:
WHAT LAWS STILL APPLY TO CHRISTIANS?
Christians do not live for the law, we live for Christ. As such, none of the laws apply. Instead, since we live for Christ, we live for the fulfillment of the law. The laws are fulfilled in us by the work of Christ in us. We obey the law of God, as He had intended it, because we are controlled by the Holy Spirit. This means it is not a work of ourselves, but a work of God.
This is no small matter. Christ made it clear in John 15:5 that apart from Him we can do nothing. This means that God values the source of our actions more than the results of our actions. If our actions are sourced from God, meaning that Jesus was the source of what we did, then we produce good fruit. However, if God is not the source of our actions, it doesn't matter what we did, we have not produced anything of value.
This means that while we live on earth, we might do mighty works. We might donate millions to charity, we might obey every law of God, and we might even help lead people to Jesus. However, if none of these actions were sourced from Christ, they are worthless. We cannot judge the success of our works based upon the results of our actions. The results are what we can see. They mean nothing. What really matters in the eyes of God: Did you do God's will? (Matthew 7:21; Matthew 12:50)
Who can possibly do God's will? First, we would have to know what God's will is. The only way to know the mind of God is to be controlled by the Spirit. The only way to produce good fruit is to be controlled by the Spirit. This means that Christ is our Lord. He rules over us. He reigns over us. He is the source of our actions. When Christ is the source of our actions, then we produce good fruit and do the will of God. Only then is our work valuable in the eyes of God because our works are God's works.
To sum up this entire post:
As Christians, we do not live by the law. We follow Christ who led us by the Holy Spirit. We live for God. The entire Word of God applies to Christians, and is still valid; however, it is interpreted and understood by the Holy Spirit. This means that we live by the guidance of the Holy Spirit alone, and not by our own understanding. We place all our confidence and trust in Christ to lead us and keep us on the path of righteousness.
I hope this helps,