This verse is sometimes understood to be saying that since keeping the law doesn’t result in justification, then the law is abolished. This interpretation creates a few problems.

First, such a view falsely assumes a purpose of God’s law was to justify. There’s no place in Scripture where it says obeying God’s law could produce justification. Instead, we always find justification to be by grace, through faith as seen in Romans 3:22, 24-28, 30 and 31. 

Next, it requires an unwarranted logical leap to say that since the law can never result in justification then it must be abolished. This broad line of reasoning needs to be thought through for its specific implications. For example, do we want to say:

1. God’s law prohibits murder.
2. Refraining from murdering does not earn justification.
3. (Logic leap) Therefore, all of God’s laws are abolished.
4. Unintended consequence: There no longer exists any objective standard to prove murder is wrong.

We need to look for an understanding that aligns with all of Scripture and that can also survive logical arguments.

Let’s get a little more immediate context for Romans 3:20.

“Now we know that whatever the law says it speaks to those who are under the law, so that every mouth may be stopped, and the whole world may be held accountable to God. For by works of the law no human being will be justified in his sight, since through the law comes knowledge of sin.” (Romans 3:19-20)

One clear purpose of the law was to define sin. Without the law, sin doesn’t exist.

“For the law brings wrath, but where there is no law there is no transgression.” (Romans 4:5)

“for sin indeed was in the world before the law was given, but sin is not counted where there is no law.” (Romans 5:13)

“What then shall we say? That the law is sin? By no means! Yet if it had not been for the law, I would not have known sin. For I would not have known what it is to covet if the law had not said, “You shall not covet.” (Romans 7:7)

“Everyone who makes a practice of sinning also practices lawlessness; sin is lawlessness.” (1 John 3:4)

To declare the law abolished is to declare sin abolished since sin is defined as breaking God’s law.

Since the law was never able to produce justification, God must have had other reasons to command obedience. Indeed, Scripture details many reasons to obey God’s law.

• One must obey to live among God’s people (Genesis 17:14, Exodus 31:14, Leviticus 17:10, Numbers 9:13, Deuteronomy 28:64) 
• Obedience is required for blessing and to avoid curses (all of Deuteronomy 28)
• Keeping God’s commandments is how we show love to Jesus and the Father (John 14:15, John 14:21, John 14:23-24, 1 John 5:3, 2 John 1:6)
• Works of obedience to the law are an outward indicator of inner faith (James 2:14-26). 
• God’s law will remain the legal code of his Kingdom when our King lives among us again. (Zechariah 14:16-21, Jeremiah 33:18, Ezekiel 45:22)
• Obedience is evidence of God’s Spirit within, according to the New Covenant. (Ezekiel 11:19-20, Ezekiel 36:26-27)
• It is for freedom (Romans 8:2).
• It is for holiness and righteousness. (Romans 7:12, 2 Peter 2:21)

While obeying the law does not produce justification, it does evidence justification. 

“For it is not the hearers of the law who are righteous before God, but the doers of the law who will be justified.” (Romans 2:13)

James also refers to “doers of the word” and clarifies that the word he’s referring to is “the perfect law, the law of liberty”.

“But be doers of the word, and not hearers only, deceiving yourselves. For if anyone is a hearer of the word and not a doer, he is like a man who looks intently at his natural face in a mirror. For he looks at himself and goes away and at once forgets what he was like. But the one who looks into the perfect law, the law of liberty, and perseveres, being no hearer who forgets but a doer who acts, he will be blessed in his doing. If anyone thinks he is religious and does not bridle his tongue but deceives his heart, this person’s religion is worthless.” (James 1:22-26)

“What good is it, my brothers, if someone says he has faith but does not have works? Can that faith save him? If a brother or sister is poorly clothed and lacking in daily food, and one of you says to them, “Go in peace, be warmed and filled,” without giving them the things needed for the body, what good is that? So also faith by itself, if it does not have works, is dead. But someone will say, “You have faith and I have works.” Show me your faith apart from your works, and I will show you my faith by my works. You believe that God is one; you do well. Even the demons believe–and shudder! Do you want to be shown, you foolish person, that faith apart from works is useless? Was not Abraham our father justified by works when he offered up his son Isaac on the altar? You see that faith was active along with his works, and faith was completed by his works; and the Scripture was fulfilled that says, “Abraham believed God, and it was counted to him as righteousness”–and he was called a friend of God. You see that a person is justified by works and not by faith alone. And in the same way was not also Rahab the prostitute justified by works when she received the messengers and sent them out by another way? For as the body apart from the spirit is dead, so also faith apart from works is dead.” (James 2:14-26)

We don’t obey to become justified. We obey because we are justified.