Do we have to follow the cleanliness laws from the Old Testament? If yes, what are they? If no, then why not?

The cleanliness laws of the Old Testament were instituted by God for several reasons: (1) to show separation from the heathen people around them. Many of the laws without being seen in that context seem absurd on their face but, with God’s desire to show Israel as being a separate, a “holy” people, the laws provided an outward, visible, tangible difference to the peoples around them; (2) in some instances, to keep them healthy. Living in the close proximity to each other that they did, the estimated two million people involved in the Exodus would have very quickly been decimated by a public health crisis, such as any kind of epidemic. Some of the laws were very much about keeping them healthy and safe; and (3) simple obedience, simply because God said so. Does there have to be a reason beyond God simply saying, “This is the way I want it?” A large part of the cleanliness laws was God attempting to get the people of Israel to understand His holiness, and to live a life within understanding how deeply pervasive God wanted His holiness to extend to them. For instance, in the book of Leviticus, the phrases “I am the Lord” and “I am holy” are used over 50 times.
Seeing as how being holy—separated—to God should seem to be an obvious desire for a Christian, do those laws still apply today? NO. Why? One reason and one reason only: Jesus Christ. The perfect holiness of Jesus Christ directly indwells each person who has surrendered their lives to Him. He has deposited Himself, in the person of the Holy Spirit, into their lives. The New Testament still calls us to holiness; the apostle Peter makes that point very plainly when he says, [W]hat kind of people ought you to be? You ought to live holy and godly lives as you look forward to the day of God and speed its coming. . . . make every effort to be found spotless, blameless and at peace with him—II Peter 3:11b-12a, 14b. And of course, Peter famously quoted from the fundamental thrust of the Mosaic Law in his first letter, where he said, As obedient children, do not conform to the evil desires you had when you lived in ignorance. But just as he who called you is holy, so be holy in all you do; for it is written, “BE HOLY, BECAUSE I AM HOLY”—I Peter 1:14-16 (quoting Leviticus 11:44, 45 and 19:2). Paul urged us, in view of God’s mercy, to offer your bodies as living sacrifices, holy and pleasing to God—this is your spiritual act of worship—Paul, Romans 12:1. Peter further identified the body of believers—the church—as a chosen people, a royal priesthood, a holy nation, a people belonging to God, that you may declare the praises of him who called you out of darkness into his wonderful light—I Peter 2:9. From the standpoint of OT holiness, Jesus fulfilled all those requirements in His own life and then, in His substitutionary death, offered the application of that fulfillment to us.
Also, in another difference, the Israelites were called by “command” to the life of holiness that reflected God—a lifestyle that was also a look ahead to the holiness that God would make available through His Son. Christ’s holiness, once given to us through His Holy Spirit, causes a response from us through love, responding to His love with the love of a dedicated life understood through personal relationship.
The point: Where once upon a time God worked through His people through His physical presence in the Tabernacle and through the physical purification rites and physical atonings of the sacrificial system, He now works with us directly—through the spiritual presence of His Son remaking us in His image.
Finally, the work of Christ was performed “once for all”; as the author of the book of Hebrews pointed out, Nor did he (Christ) enter heaven to offer himself again and again, the way the high priest enters the Most Holy Place every year with blood that is not his own. Then Christ would have had to suffer many times since the creation of the world. But now he has appeared once for all at the end of the ages to do away with sin by the sacrifice of himself—Hebrews 9:25-26. In the world of the Old Testament, the Israelites were perpetually needing to perform their ritual sacrifices (Yom Kippur—the Day of Atonement, an “annual” event) and continually undergoing ritual cleansings of all sorts. In the world of the New Testament, I have been purified once and forever by the cleansing sacrifice of Christ Himself. I am now ALWAYS seen by God through the filter of His Son in my life—a condition that will ALWAYS be present in me. As Christ Himself said, My sheep listen to my voice; I know them, and they follow me. I give them eternal life, and they shall never perish; no one can snatch them out of my hand. My Father, who has given them to me, is greater than all; no one can snatch them out of my Father’s hand—Jesus Christ, John 10:27-29. And as Paul so wonderfully reiterated to the church at Corinth, Now it is God who makes both us and you stand firm in Christ. He anointed us, set his seal of ownership on us, and put his Spirit in our hearts as a deposit, guaranteeing what is to come—Paul, II Corinthians 1:21-22.