What is the most common misconception surrounding the New Covenant?

Some misconceptions:

There are people that think that Jesus gives a license to sin, because He removes all of our sins.

That the New Covenant renders all of God's laws in the Old Covenant obsolete. Yet God's righteousness and His desires do not change. What God hated back then, He still hates today. With the New Covenant, we have the permanent infilling of the Holy Spirit, so we are not like people trying to obey the laws of a God who feels foreign to us - but His laws are now close to our heart, even being within us. Our connection to God's Holy Spirit, impresses His righteous desires upon our hearts. So what was once unnatural to us, is now natural.

The Old Covenant was man trying to be obey the letter of the law, but failing because his heart was not right. But Jesus kept the law unto perfection, and gives us His own Spirit, that same Spirit which overcomes the world, even subduing the infirmities within us - with His Righteousness piercing through the darkness. He is the Source of Goodness, Truth, Love, Life, and Righteousness. When we are separated from Him, we abide in darkness and unrighteousness - and we are avoid. He is our completion, and we need Him to be whole.

This one is not the most common:

That the New Covenant is for the Gentiles, but the Old Covenant is for the Jews -> I believe John Hagee teaches this. It's called dual covenant theology. Supposedly Jerry Falwell also, but I haven't looked into him. Jesus is the Savior of His people, to the believing Jews and the believing Gentiles. The flesh profits nothing. No one can reject God and have Him simultaneously. If we reject Jesus, we reject God, because Jesus is the revelation of Him, the image of the invisible God -- the Spirit (God Himself) made manifest in the flesh of a man.

God doesn't give us a license to sin. The Holy Spirit filled Christian naturally desires to obey God's laws, not in order to work to gain salvation, but because he/she is saved - and God is working in the person, perfecting the person. Spiritual growth is God's way, not remaining dead in our sins, living like their is a chasm between us and our Righteous Creator -- we now have access to Him - through the bridge and mediator Jesus.

These two responders got that correct:

"the one which gives folks a false sense of security that they are saved"

"Jesus gives a license to sin, because He removes all of our sins."

I don't know where these beliefs came from. They aren't even Christian.

We sin all our lives, meaning we have to keep repenting, and resolving to do better. Jesus' sacrifice was not a get-out-of-hell-free card. We still have to lead a Christian life, which includes repentance. And even then, we are given salvation -- after we die -- only by the grace of God.

That Saul of Tarsus was an Apostle. Not in the letters in Red.

That keeping the original Ten Commandments or any of the 600+ corresponding statutes and ordinances, will somehow, get you into heaven. And that the Old Covenant still exists for some people, somehow (which it does not). Also, in spite of the promises and personal guarantees furnished by Jesus Christ, that the Holy Catholic Church he founded, for the purpose of our salvation, secretly apostatized at some mysterious point in time, and became something else, entirely (also not true).

The most common misconception around the New Covenant in my humble opinion is the one which gives folks a false sense of security that they are saved, if they try not to swear, smoke or drink and they try to be good and do good deeds for people.

This is earning their own salvation which is impossible because if it were possible to get to heaven and be saved by our own efforts and good deeds, then Christ died for nothing.

The only way for anyone of us human beings to be forgiven of our sins and change from being a sinner to being saved by grace (that's undeserved or unmerited favour) is to believe in our hearts the Lord Jesus and confess with our mouths that God has raised Him from the Dead, for with the heart one believes unto righteousness and with the mouth confession is made unto salvation (Romans 10: 9 & 10).

Jesus Christ is the only Saviour of the World, because He is the only sinless, blameless spotless Lamb of God who takes away the sin of the world and Jesus died on the Cross in order that we could be saved if we turn to Him, repent of our sins and follow and obey Him for life because Jesus is the only Name under heaven given amongst men were by we Must be saved (Acts 4:12) and the only WAY to the Father in heaven. (John 14:6)

That all good people or all Christians go to heaven. Only a small number, a little flock, are actually in the covenant that Jesus made with his apostles for a kingdom, that is, are chosen to serve in heaven.

That any of it happened.

The Old Testament is still true.

That vegetarianism would no longer be favorable to God.

That we are no longer under the laws of The Old Testament.