What is the new covenant of Hebrews 10, that God will write in the hearts of His people, and He will wipe out their sins?
The passage to which you refer is in Hebrews 10:16-17, and follows a discussion of two previous covenants.
At the beginning of the chapter, the author speaks of “the law” which could “never with these same sacrifices, which they offer continually year by year make those who approach perfect.” He speaks of the sacrifices required under the Mosaic system, where the blood of bulls and of goats was offered for transgressions but could not remove sin (Heb 10:4). Then, by way of contrast, he mentions a better system than that of sacrifices and offerings. It is the offering of one’s self in obedience to the will of God, as Christ did: “Then I said, ‘Behold, I have come- In the volume of the book it is written of Me- To do Your will, O God’ ” (v. 7). He then comments that the sacrifices and offerings under the law did not give pleasure to God as did the law of obedience (v. 8), then adds that the first (the law of sacrifice and offering) was taken away to make place for the second (the offering of one’s self in obedience). The latter was the sacrifice which Jesus offered, in contrast to the daily offerings of the priests under the law, “which can never take away sins” (10:4). The second plan, which Christ followed, was the far more effective, after which He “sat down at the right hand of God, from that time waiting till His enemies are made His footstool” (Heb. 10:12-13). The second plan, of offering our bodies a living sacrifice, wholly consecrated to God, is the arrangement by which all who would partake of the blessings with Christ must be “sanctified” (Heb. 10:14).
Verse 15 then speaks of a third covenant, which applies after the two previous. We read: “after that he had said before,” and the passage quoted is from Jeremiah 31:33-34: “this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put My law in their minds, and write it on their hearts … I will forgive their iniquity, and their sin I will remember no more.” Then the final comment is added, “Now where there is remission of these, there is no longer an offering for sin” (Heb 10:18). That is, where sins are forgiven, there is no more need for any sacrifice.
In Jeremiah 31, the context of this passage (vs. 33-34) indicates the covenant arrangement under which the populace of the new kingdom set up by Jesus at His return will live. They, too, like those who went before them, will have to be purified, tried, developed in character, so as to qualify for a favored place in the Divine plan (Zech. 13:9). Ezekiel speaks also of this refining and purifying, as a covenant: “I will make you pass under the rod, and I will bring you into the bond of the covenant; I will purge the rebels from among you, and those who transgress against Me; I will bring them out of the country where they dwell, but they shall not enter the land of Israel. Then you will know that I am the Lord” (Ezek. 20:37-38). The rod indicates law enforced, as in Rev. 2:26-27, speaking of Christ, “He shall rule them with a rod of iron.” The purging indicates the removing of those who rebel, who will not accept the authority of God’s law.
Once the purifying and purging are complete and the rebellious element is removed, those who remain will be those submissive to law, and these will be willing to walk in God’s ways. Of them it will be written, “I will forgive their iniquity, and I will remember their sin no more” (Jer. 31:34). They are a generation of people instructed in the ways of the Lord, as the same passage indicates: “They shall teach no more every man his neighbor, and every man his brother, saying, Know the Lord: for they shall all know me, from the least of them unto the greatest of them, saith the Lord.” And they will also be a people obedient. God promises forgiveness to all who turn from their evil ways (Ezek. 18:27; 33:18-19). So abundant is God’s forgiveness, that He promises not even to mention the sins of those who have turned and done “that which is lawful and right” (Ezek. 33:16).
The result will be a people and a time very different from that familiar to us now, when sin and lawlessness are rampant and the innocent often have to suffer.
And when there is no more sin, there will not need to be any more “offering for sin” (Heb. 10:18).
At the end of the Millennium, when everyone who lives will be immortal and earth becomes a part of heaven (Matt. 6:10); all sacrificing, including the sacrificing of oneself as an offering to God, will cease.