About what law is Jeremiah 31:33 speaking?
The clue for resolving this question [i.e., the question found in the title of this post] is found in the contrast and parallel between the Old and New Covenants stated in these verses (cf. vv. 32, 33a, "not like the covenant which I made with their fathers... But this is the covenant which I will make..."). Clearly, there is a contrast in these verses between the Old and New Covenants. But that very contrast assumes and implies a parallel. Let me state the contrast clearly. The Old Covenant was broken because God wrote His law on stone and not on all the hearts of His people. The New Covenant will not be broken, because God will write His law on the hearts of all His covenant people.
The clear contrast here is the place where the law is written. In the Old Covenant, the place is on stone tablets. In the New, it is the fleshy heart. But in this contrast there is also clearly a parallel. In both covenants, God writes His law. The contrast clearly assumes and implies the parallel. The contrast in where the law is written, however, assumes that the law under discussion still has a vital place to play in God's New Covenant.
In light of this clear parallel, we may return to our question with a better understanding of its answer. About what law is verse 33 speaking? Two things clearly identify this law.
First, it is the law written by God Himself and by His own finger. This is clear from verse 33, "I will put my law within them, and on their heart I will write it..." But the only law so written was the Moral Law of God as summarized in the Ten Commandments. It is the Ten Commandments, and those Ten Commandments alone, which were written by God Himself with His own finger.
[The authors quote Exodus 24:12; 31:18; 32:16; 34:1 and Deuteronomy 10:1-4 to prove that God personally wrote the Ten Commandments with His own finger. One thing that I found interesting about these quotes: for some reason I was under the impression that after Moses destroyed the first tablets of the Ten Commandments in his anger, he had carved the second copy himself, but the text is very clear that God directly wrote the second copy of the Ten Commandments, just as He had written the first.]
Other aspects of the Old Covenant law, the Judicial and Ceremonial, were written, not by God Himself, but by Moses. "And Moses wrote down all the words of the LORD" (Exo 24:4; cf. 34:10-27).
Second, it is the law written on stone that is re-written in the New Covenant on the heart of all covenant participants. The emphasis on the place where God's law is written in Jeremiah 31:33 plainly suggests this thought. This is confirmed by the references of the Apostle Paul to this verse in 2 Corinthians 3:1-8. Here Paul uses the very words to speak of the stone tablets in the Septuagint (the Greek translation of the Hebrew Scriptures [LXX]) of Exodus 31:18 and 34:4. The Judicial Law of Israel was not written on stone, but in a book (Exo 24:3, 4, 7; contrast these with v. 12). The Ceremonial Law of Israel was not written on the heart. On the Moral Law, as epitomized and summarily contained in the Ten Commandments, was written on stone.