The covenants God made in the OT with the Hebrews were “for ever.” However, from what we understand in the book of Hebrews, they could only continue if the priesthood continued, and if God Himself continued it. As it is, the Aaronic priesthood has been replaced by Jesus’ Melchizedec priesthood (Hebrews 7), and God Himself rejected the law because it never did what He wanted; the old covenant was replaced by the new covenant of faith in Christ (Hebrews 8:7-13). God said the OT law was ineffective and ready to get the boot. Why keep something around that never perfected anyone (Heb. 10:1)?
Yet in Matthew 5:18 Jesus said the law will continue though heaven and earth pass away. One of the ways the law continues in a practical way is found in Romans 3 and 7. There, we learn that we cannot be made righteous before God through the law, but through the law we learn that we are sinners (3:20). So this rules out any possibility of following the OT laws in order to be righteous before God, or to be saved.
Romans 7 sheds more light on this. Paul the apostle thought he would be righteous because he kept the law. However, he found that the law condemned the thoughts and intents of his heart (Rom. 7:18-20). He wants to do the law, but finds another law in him that prevents him from obeying the law, the law of sin in him. For instance, we may follow the law and do good, but inside we become proud and think we’re better than anyone else, and the law exposes our hypocrisy. This is the correct use of the law: it is to turn us away from trusting in our works or our self to depend completely on Christ (Rom. 7:24-25; also see Galatians 3:24-25).
First Tim. 1:5-11 gives us more lessons on how to use the law. It is not for those who are living by faith, but for those who are living unrighteously and need to be convicted of sin. Whether you show the 10 Commandments to an unsaved neighbor or a backslidden Christian, the effect would be the same: to convict them that they are not living by faith in Christ, but trusting that their own way is good enough. To break the law is to be guilty before God and in need of forgiveness, and repentance that results in living by faith in Christ.
James 2:8-13 is very clear: if we try to keep the law, we have to keep all of it, including the animal sacrifices and priesthood; if we fail to keep any of it, we are condemned as though we had broken all the law. On the other hand, if we keep the law of love, and show mercy to others, then at the judgment God will show love and mercy to us, and wipe out all remembrance of sin. (Showing a lack of mercy to your neighbor will mean receiving a lack of mercy from God: Matthew 6:14-15, 18:23-35.)
I hope this shows you that though the law was fulfilled by Christ, it continues in a different way – to point us to Christ. If we live by faith, then that law is fulfilled in us too: Romans 8:4!