The entire book of Hebrews explains that the Old Testament ceremonial laws were not so much abolished as fulfilled by Christ. Whenever we pray ‘in Jesus’ name,’ we ‘have confidence to enter the Most Holy Place by the blood of Jesus’ (Hebrews 10:19).
It would, therefore, be deeply inconsistent with the teaching of the Bible as a whole if we were to continue to follow the ceremonial laws.
The New Testament gives us further guidance about how to read the Old Testament.
Paul makes it clear in places like Romans 13:8ff that the apostles understood the Old Testament moral law to still be binding on us. In short, the coming of Christ changed how we worship but not how we live.
The moral law is an outline of God’s own character—his integrity, love and faithfulness. And so all the Old Testament says about loving our neighbor, caring for the poor, generosity with our possessions, social relationships and commitment to our family is still in force. The New Testament continues to forbid killing or committing adultery, and all the sex ethics of the Old Testament are restated throughout the New Testament (Matthew 5:27-30; 1 Corinthians 6:9-20; 1 Timothy 1:8-11).
If the New Testament has reaffirmed a commandment, then it is still in force for us today.
Further, the New Testament explains another change between the Testaments.
Sins continue to be sins—but the penalties change. In the Old Testament, things like adultery or incest were punishable with civil sanctions like execution. This is because at that time God’s people existed in the form of a nation-state and so all sins had civil penalties.
But in the New Testament, the people of God are an assembly of churches all over the world, living under many different governments.
The church is not a civil government, and so sins are dealt with by exhortation and, at worst, exclusion from membership. This is how a case of incest in the Corinthian church is dealt with by Paul (1 Corinthians 5:1ff and 2 Corinthians 2:7-11).
Why this change?
Under Christ, the gospel is not confined to a single nation—it has been released to go into all cultures and peoples.
Once you grant the main premise of the Bible—about the surpassing significance of Christ and his salvation—then all the various parts of the Bible make sense.