What in New Testament invalidates God’s commandments from the Old Testament?

To me it is not a matter of invalidation, but more the fact that Jesus Christ is a fulfillment of the Old Testament laws. He made it perfectly clear when asked a similar question:

37 Jesus said unto him, “Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. 38 This is the first and great commandment. 39 And the second is like unto it, Thou shalt love thy neighbour as thyself. 40 On these two commandments hang all the law and the prophets.” - Matthew 22

All of the Ten Commandments are still in force. Jesus discussed many of them in Matthew 5:17-48 and explicitly upheld them:

17 “Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. 18 For truly I tell you, until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished. 19 Therefore anyone who sets aside one of the least of these commands and teaches others accordingly will be called least in the kingdom of heaven, but whoever practices and teaches these commands will be called great in the kingdom of heaven. 20 For I tell you that unless your righteousness surpasses that of the Pharisees and the teachers of the law, you will certainly not enter the kingdom of heaven.

_{Matthew 5:17-20 (NIV)}

Not only did Jesus uphold the Ten Commandments but He actually made them stronger. For example, not only must you avoid a physical act of adultery but you cannot even look at a person with lust.

However, the Jewish ceremonial laws were abolished. Some examples of ceremonial laws that are no longer in force are given in another answer. Another example regards impure or unclean food (defined in Leviticus 11); the apostle Peter had a vision in which he was told that no food was to be considered impure:

9 About noon the following day as they were on their journey and approaching the city, Peter went up on the roof to pray. 10 He became hungry and wanted something to eat, and while the meal was being prepared, he fell into a trance. 11 He saw heaven opened and something like a large sheet being let down to earth by its four corners. 12 It contained all kinds of four-footed animals, as well as reptiles and birds. 13 Then a voice told him, “Get up, Peter. Kill and eat.”

14 “Surely not, Lord!” Peter replied. “I have never eaten anything impure or unclean.”

15 The voice spoke to him a second time, “Do not call anything impure that God has made clean.”

16 This happened three times, and immediately the sheet was taken back to heaven.

_{Acts 10:9-16 (NIV)}

Jesus Himself alluded to the fact that food does not "defile" a person but that evil things which come out from the heart do:

16 “Are you still so dull?” Jesus asked them. 17 “Don’t you see that whatever enters the mouth goes into the stomach and then out of the body? 18 But the things that come out of a person’s mouth come from the heart, and these defile them. 19 For out of the heart come evil thoughts—murder, adultery, sexual immorality, theft, false testimony, slander. 20 These are what defile a person; but eating with unwashed hands does not defile them.”

_{Matthew 15:16-20 (NIV)}

Notice that all the "evil thoughts" which Jesus condemned correspond to things forbidden by the Ten Commandments.

The Letter to the Hebrews also discusses the contrast between the old covenant and the new. It mentions that "ceremonial" regulations do not apply in the new covenant. It also explains that Jesus serves as the high priest who offers a sacrifice of His own blood in Heaven itself and once and for all time, as opposed to the Jewish high priest offering a sacrifice of an animal's blood in an earthly tabernacle once every year:

Now the first covenant had regulations for worship and also an earthly sanctuary. 2 A tabernacle was set up. In its first room...

6 ...the priests entered regularly into the outer room to carry on their ministry. 7 But only the high priest entered the inner room, and that only once a year, and never without blood, which he offered for himself and for the sins the people had committed in ignorance. 8 The Holy Spirit was showing by this that the way into the Most Holy Place had not yet been disclosed as long as the first tabernacle was still functioning. 9 This is an illustration for the present time, indicating that the gifts and sacrifices being offered were not able to clear the conscience of the worshiper. 10 They are only a matter of food and drink and various ceremonial washings—external regulations applying until the time of the new order.

11 But when Christ came as high priest of the good things that are now already here, he went through the greater and more perfect tabernacle that is not made with human hands, that is to say, is not a part of this creation. 12 He did not enter by means of the blood of goats and calves; but he entered the Most Holy Place once for all by his own blood, thus obtaining eternal redemption. 13 The blood of goats and bulls and the ashes of a heifer sprinkled on those who are ceremonially unclean sanctify them so that they are outwardly clean. 14 How much more, then, will the blood of Christ, who through the eternal Spirit offered himself unblemished to God, cleanse our consciences from acts that lead to death, so that we may serve the living God!

15 For this reason Christ is the mediator of a new covenant, that those who are called may receive the promised eternal inheritance—now that he has died as a ransom to set them free from the sins committed under the first covenant.

...

24 For Christ did not enter a sanctuary made with human hands that was only a copy of the true one; he entered heaven itself, now to appear for us in God’s presence. 25 Nor did he 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. 26 Otherwise Christ would have had to suffer many times since the creation of the world. But he has appeared once for all at the culmination of the ages to do away with sin by the sacrifice of himself. 27 Just as people are destined to die once, and after that to face judgment, 28 so Christ was sacrificed once to take away the sins of many; and he will appear a second time, not to bear sin, but to bring salvation to those who are waiting for him.

_{Hebrews 9}

There's a simpler argument, I think, than looking at the text of the New Testament to determine if the Old Testament laws apply (though the support from the NT is good). We can simply ask: Between which parties was the Mosaic covenant made?

It should be pretty clear that the Mosaic covenant was made between the Jews and God. Exodus 24:8 says,

Moses then took the blood, sprinkled it on the people and said, “This is the blood of the covenant that the Lord has made with you in accordance with all these words.”

If, then, that covenant is between God and the Jews, why would we be bound to obey it? Obviously we wouldn't. The question is a little more complicated for Jewish Christians, which is where the NT passages become important (but the other answers cover that well).

Greg Koukl, an apologist for Stand to Reason, gives an illustration roughly like this:

It's illegal to murder someone in Illinois. Does that make it illegal to murder someone in California? No! It's illegal in both places because both places have laws that reflect a higher principal: murder is wrong. In the same way, we are held to portions of the laws in the OT not in virtue of their appearing in the OT but because they reflect higher moral truths.

Let us suppose that we wanted to keep all the commandments in the Torah. If we intended to do that, then we would need:

• A temple in Jerusalem
• All people that believed this would need to attend Jerusalem three times per year for the annual feasts
• We would need a Levitical priesthood
• We would need a high priest descended from Aaron
• We would also need a ready supply of animals for sacrifice
• sacred vessels for the numerous ceremonies
• etc.

None of this is available! In any case, the Jerusalem council (Acts 15) defined clearly what was need of Christians - to keep the moral law and not eat blood.

The whole letter to the Galatians is evidence the law has ceased in its ceremonial aspect, and the moral law has ceased in its condemning aspect for those who truly believe.

But now, after that ye have known God, or rather are known of God, how turn ye again to the weak and beggarly elements, whereunto ye desire again to be in bondage? 10 Ye observe days, and months, and times, and years. 11 I am afraid of you, lest I have bestowed upon you labour in vain. Galatians 4:9-11.

14 Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross; 15 And having spoiled principalities and powers, he made a shew of them openly, triumphing over them in it. 16 Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days: 17 Which are a shadow of things to come; but the body is of Christ. Colossians 2:14-17

7 Know ye not, brethren, (for I speak to them that know the law,) how that the law hath dominion over a man as long as he liveth? 2 For the woman which hath an husband is bound by the law to her husband so long as he liveth; but if the husband be dead, she is loosed from the law of her husband. 3 So then if, while her husband liveth, she be married to another man, she shall be called an adulteress: but if her husband be dead, she is free from that law; so that she is no adulteress, though she be married to another man. 4 Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God. 5 For when we were in the flesh, the motions of sins, which were by the law, did work in our members to bring forth fruit unto death. 6 But now we are delivered from the law, that being dead wherein we were held; that we should serve in newness of spirit, and not in the oldness of the letter. Romans 7:1-6

And the Apostle Paul's argument arrives at these verse in chapter 8:

8 There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit. 2 For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. 3 For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: 4 That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit. Romans 8:1-4

Those who try to get right with God by trying to keep the Old Testament law will only ever feel condemnation and the law stirring up in them the very sins they are trying to destroy. They will then either be forced to give up, or they will start justifying themselves by imagining that the law only requires an external obedience.

But faith in Jesus is the ground of our acceptance with our Father. This leads us to accepting that the moral law is spiritual and fully to be obeyed down to the very heart, only we cannot keep it, so daily we pray forgive us our trespasses, and are not condemmed. The condemning aspect of the moral law has no force over the believer in Christ, he is fully dead to the law in this sense of condemnation (Romans 7:4 and 8:1).

Pretty much the whole letter to the Hebrews is evidence that the Old Covenant or Old Testament way has finished and been replaced by a New Covenant or New Testament. The Old Testament was imperfect and could not make perfect those who were under it, and so there needed to be continual sacrifices made. But the under the New Testament there only needed to be one sacrifice for all time, because the sacrifice was perfect and makes perfect by faith those who believe in it.

Not sure how you can imagine that the Old Testament ceremonial law is still in force, when so much of the New Testament is plainly saying it isn't.

For more on this topic see the answer here: https://hermeneutics.stackexchange.com/questions/40135/if-we-died-to-the-law-disannulling-of-the-commandment-took-place-the-why-i/40152#40152

The law serves a good and honourable function - it is a schoolmaster to bring to Christ. It teaches us of our need of him. And then, it has done its work.

Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. [Galatians 3:24, KJV.]

And then, if we believe, we are dead to it :

Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God. {Romans 7:4, KJV.]

The 'ceremonies' are the sacrificial artefacts which pre-figure the coming of Christ and the offering up of himself. They are not 'law' as such, they were an arrangement with Israel in order to set forth the gospel.

Law - the ten commandments engraved on stone - is absolute.

And this was nailed to the cross, by God himself :

Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to the cross; [Colossians 2:14, KJV, but correctly stating 'the' cross, not 'his cross'.]

The Old Testament contains many commandments.

Some of them the New Testament explicitly says are still in force. In Jesus' Sermon on the Mount he addresses some of these, including murder, adultery, and covetousness.

Some of them the New Testament explicitly says are obsolete. That includes all regulations surrounding temple worship. As Jesus told the woman at the well in John 4:

21 “Woman,” Jesus replied, “believe me, a time is coming when you will worship the Father neither on this mountain nor in Jerusalem. 22 You Samaritans worship what you do not know; we worship what we do know, for salvation is from the Jews. 23 Yet a time is coming and has now come when the true worshipers will worship the Father in the Spirit and in truth, for they are the kind of worshipers the Father seeks. 24 God is spirit, and his worshipers must worship in the Spirit and in truth.”

The entire Book of Hebrews speaks of the old sacrificial system and how it really pointed to Jesus and his death on the cross. Since Jesus has paid the price once and for all time, no more blood sacrifices are necessary.

Another regulation concerns circumcision. The early church argued fiercely about this, but the Apostle Paul confronted the church leadership over the issue and they agreed that Gentile believers did not need to be circumcised. You can read about this in Acts 15:

6 The apostles and elders met to consider this question. 7 After much discussion, Peter got up and addressed them: “Brothers, you know that some time ago God made a choice among you that the Gentiles might hear from my lips the message of the gospel and believe. 8 God, who knows the heart, showed that he accepted them by giving the Holy Spirit to them, just as he did to us. 9 He did not discriminate between us and them, for he purified their hearts by faith. 10 Now then, why do you try to test God by putting on the necks of Gentiles a yoke that neither we nor our ancestors have been able to bear? 11 No! We believe it is through the grace of our Lord Jesus that we are saved, just as they are.”

...

19 “It is my judgment, therefore, that we should not make it difficult for the Gentiles who are turning to God. 20 Instead we should write to them, telling them to abstain from food polluted by idols, from sexual immorality, from the meat of strangled animals and from blood. 21 For the law of Moses has been preached in every city from the earliest times and is read in the synagogues on every Sabbath.”

The latter part of the quote gives the highlights of what parts of the Law were to remain in force: no idol worship or participation in pagan rituals, humane slaughter of animals, and no sexual immorality.

The Apostle Paul also spoke to the issue of celebrating the Jewish holidays, saying that they were not obligatory. In Galatians 4 he says:

8 Formerly, when you did not know God, you were slaves to those who by nature are not gods. 9 But now that you know God—or rather are known by God—how is it that you are turning back to those weak and miserable forces[d]? Do you wish to be enslaved by them all over again? 10 You are observing special days and months and seasons and years! 11 I fear for you, that somehow I have wasted my efforts on you.

And in Romans 14, Paul says:

5 One person considers one day more sacred than another; another considers every day alike. Each of them should be fully convinced in their own mind. 6 Whoever regards one day as special does so to the Lord.

This still leaves many parts of the Old Testament Law that were never rescinded nor commanded. Christians have been debating those rules ever since. However, if you read the New Testament carefully, you will at least find that the Ten Commandments are all still in force, since at one point or another all of them are covered.

However, even those commandments that are still in force are to be looked at in a new light. You are not justified by following them. Instead, having trusted in Christ by faith, as a saved person under the instruction of the Holy Spirit, you will desire to follow them out of love.