Many denominations believe that Christians should obey the ten commandments (also known as the decalogue), which can be found in Exodus 20:1-17. However, there are 613 commandments in the Old Testament, not just 10. Moreover, Jesus himself claimed in Matthew 22:36-40 that the two greatest commandments are not part of the decalogue. In other words, Jesus endorsed the observance of two non-decalogue commandments. This makes me wonder: how many of the 603 non-decalogue commandments are still morally binding for Christians today? I know that the two greatest commandments definitely are, but are there others?
Jesus affirmed the two greatest commandments because "All the Law and the Prophets hang on these two commandments" (Matthew 22:40). Paul says similarly in Romans 13:8, "Let no debt remain outstanding, except the continuing debt to love one another, for whoever loves others has fulfilled the law".
The Law, especially the Decalogue, was given as an expression of the two greatest commandments - to love God and neighbour.
The other 613 laws are more complicated. It's common to speak of the tripartate division of the law - moral, civil, and ceremonial. As that link shows there are complexities. Some commands are specifically abrogated in the New Testament (e.g. Mark 7:19). Jesus fulfilled the need for sacrifices to be made (Hebrews 7:18-19).
I would suggest a more nuanced approach than seeing it as a binary thing (commands we either do or do not need to obey). I think the commands are there to teach us how to live, but not to be obeyed in the Old Testament sense. This is because in the New Testament we need to walk in step with the Spirit, who alone can give us a deeper obedience to the Law. Galatians 3:24, "So the law was our guardian until Christ came that we might be justified by faith."
Jesus came to fulfill the law (Matthew 5:17). This means that the righteous demands of the law can only be fulfilled in us through him, through his death and resurrection. We cannot obey perfectly, yet through Christ and the Holy Spirit working in us we can begin to love as God wants us to love. Rather than being an external obedience, God writes his law on our hearts (Jeremiah 31:33).
Therefore the law that needs to be obeyed is the law of love - the two greatest commandments. But the rest of the law we need to see through the lens of being fulfilled in Jesus Christ.
How many non-decalogue commandments from the Old Testament are still morally binding for Christians?
Do not know how many there are, but there still exists a few here and there.
Here are six Bible passages that deal directly with bestiality or zoophilia (sexual contact with animals):
Exodus 22:19 Whosoever lieth with a beast shall surely be put to death.
Leviticus 18:23 Neither shalt thou lie with any beast to defile thyself therewith: neither shall any woman stand before a beast to lie down thereto: it is confusion.
Leviticus 20:15 And if a man lie with a beast, he shall surely be put to death: and ye shall slay the beast.
Leviticus 20:16 And if a woman approach unto any beast, and lie down thereto, thou shalt kill the woman, and the beast: they shall surely be put to death; their blood shall be upon them.
Deuteronomy 27:21 Cursed be he that lieth with any manner of beast. And all the people shall say, Amen.
Jude 1:7 Even as Sodom and Gomorrha, and the cities about them in like manner, giving themselves over to fornication, and going after strange flesh, are set forth for an example, suffering the vengeance of eternal fire.
As we can see, God clearly abhors any type of sexual contact with animals. It goes against God’s design for procreation and sexual behavior. As Leviticus 18:23 pointed out, this is nothing more than confusion!
God’s Word clearly tells us that sexual immorality of any kind is a vile sin, and unless a person repents of such wickedness, he or she will not inherit the kingdom of God:
Know ye not that the unrighteous shall not inherit the kingdom of God? Be not deceived: neither fornicators, nor idolaters, nor adulterers, nor effeminate, nor abusers of themselves with mankind, nor thieves, nor covetous, nor drunkards, nor revilers, nor extortioners, shall inherit the kingdom of God” (1 Corinthians 6:9-10).
What non-decalogue commandments from the Old Testament are still morally binding for Christians?
- Leviticus 18:16 in regards to Mark 6:17-18 :
Mark 6:17-18 [NIV]
 For Herod himself had given orders to have John arrested, and he had him bound and put in prison. He did this because of Herodias, his brother Philip’s wife, whom he had married.  For John had been saying to Herod, “It is not lawful for you to have your brother’s wife.”
Leviticus 18:16 [MT]
You shall not uncover the nakedness of your brother's wife. It is your brother's nakedness. (עֶרְוַ֥ת אֵֽשֶׁת־אָחִ֖יךָ לֹ֣א תְגַלֵּ֑ה עֶרְוַ֥ת אָחִ֖יךָ הִֽוא)
- Leviticus 18:22 in regards to Mark 10:6-8.
Mark 10:6-9 [NIV]
 "But at the beginning of creation God made them male and female."  "For this reason a man will leave his father and mother and be united to his wife,"  "and the two will become one flesh. So they are no longer two, but one flesh."
Leviticus 18:22 [MT]
You shall not lie down with a male, as with a woman. This is an abomination. (וְאֶ֨ת־זָכָ֔ר לֹ֥א תִשְׁכַּ֖ב מִשְׁכְּבֵ֣י אִשָּׁ֑ה תּֽוֹעֵבָ֖ה הִֽוא)
- Leviticus 26:1 in regards to Mark 12:15-17.
Mark 12:15-17 [NIV]
But Jesus knew their hypocrisy. “Why are you trying to trap me?” he asked. “Bring me a denarius and let me look at it.”  They brought the coin, and he asked them, “Whose image is this? And whose inscription?” - “Caesar’s,” they replied.  Then Jesus said to them, “Give back to Caesar what is Caesar’s and to God what is God’s.”
Leviticus 26:1 [MT]
**You shall not make idols for yourselves, nor shall you set up a statue or a monument for yourselves**. And in your land you shall not place a pavement stone on which to prostrate yourselves, for I am YHVH, your-God.(לֹא־תַֽעֲשׂ֨וּ לָכֶ֜ם אֱלִילִ֗ם וּפֶ֤סֶל וּמַצֵּבָה֙ לֹֽא־תָקִ֣ימוּ לָכֶ֔ם וְאֶ֣בֶן מַשְׂכִּ֗ית לֹ֤א תִתְּנוּ֙ בְּאַרְצְכֶ֔ם לְהִשְׁתַּֽחֲוֹ֖ת עָלֶ֑יהָ כִּ֛י אֲנִ֥י יְהֹוָ֖ה אֱלֹֽהֵיכֶֽם)--------------
- Leviticus 19:17 in regards to Mark 11:25.
Mark 11:25 [NIV]
And when you stand praying, if you hold anything against anyone, forgive them, so that your Father in heaven may forgive you your sins.
Leviticus 19:17 [MT]
You shall not hate your brother in your heart. You shall surely rebuke your fellow, but you shall not bear a sin on his account. (לֹֽא־תִשְׂנָ֥א אֶת־אָחִ֖יךָ בִּלְבָבֶ֑ךָ הוֹכֵ֤חַ תּוֹכִ֨יחַ֙ אֶת־עֲמִיתֶ֔ךָ וְלֹֽא־תִשָּׂ֥א עָלָ֖יו חֵֽטְא)