Here's the passage:
Does this mean that we should never discuss interpretations of the law? Does this mean that differing interpretations of the law should never cause Christian brothers to separate from one another? I will show that both answers are "no." What does Paul mean, then?
Divisions in the early church
Paul was in a difficult position as an early church leader. Most of the early converts to Christianity were Jews. Their entire worldview was shaped by a group of Biblical laws which we'll call "Jew/Gentile separation laws." These included circumcision requirements, prohibitions on intermarriage, food laws, ritual "cleanness" laws, and some others. These laws had an important purpose in Israelite history, which was to prevent God's people from syncretizing (mixing culturally/religiously) with the surrounding Gentile nations. God made this purpose clear in the law; for example:
When Jesus established the New Covenant, he opened the way for Gentiles to have the same close relationship with God that the Jews had as God's chosen people:
Most of the "growing pains" of the early church came from integrating Gentiles into that previously "Jewish-only" relationship. Gentiles had a variety of cultural practices that they would have to abandon (see Acts 15), and Jews did too, in the form of those laws which were separating them from the Gentiles:
Much of the time, when Paul spoke of "the law," he was referring to this set of ceremonial and separation laws, which do not apply in the New Covenant.
The rise of the Judaizers
Many Jews who wanted to follow Christ could not accept this new teaching. They had grown up with hostility toward the Gentiles, and now Jesus was commanding them to reconcile. Many could not, and believed that Gentiles had to be circumcised, enter into the Mosaic Covenant, and follow all the ritual cleanness laws, food laws, feast days, etc. These people were the Judaizers, and even the Apostle Peter was influence by them. Paul had to confront him publicly:
When Paul said "avoid ... quarrels about the law," he was referring to people who rejected the New Covenant teaching that the Gentiles could freely enter without becoming Jews. The presupposition of the Judaizers was that all the Mosaic laws (including the Jew/Gentile separation laws) were still in effect, even under the New Covenant. If this is your presupposition, then you will naturally reject Paul's teaching, in a variety of places.
Apologetics pro-tip: Don't debate presuppositions
Paul is exactly right: disputes over presuppositions are "unprofitable and worthless." Neither person will ever convince the other, so the discussion is not only a waste of time, it is likely to lead to "raised emotions" on the part of one or the other.
Here's the paradox: once you both understand that your positions are logically irreconcilable, that should actually ease a lot of the tension. Unless you think that it is right to force your presuppositions on the other person (and you won't, if you are a follower of Christ: Matt. 20:25), then he shouldn't feel threatened by you. There are plenty of non-Christians who want to force their views on Christians by wielding "democracy" and the coercion of civil government. No true Christian should do this.
Iron sharpens iron; but iron, attempting to cut stone, will merely dull the blade and chip the stone. Presuppositions are the bedrock of epistemology, and you will just dull your blade on them.
Discussions between people who have shared presuppositions can be fruitful. It's perfectly fine to discuss (and even argue about) interpretations of the law, as long as it leads to greater understanding by one or both persons. But if it seems to be leading to a quarrel, then you're done. You failed to detect that you were hitting the bedrock of a presupposition. Cut your loss and move on to discuss something that won't cause animosity.
Separation over the law
Does Paul's statement mean that we should never separate from other brethren over interpretations of Biblical law? Absolutely not. Let's pick a nice, controversial one: polygyny (the practice of a husband having more than one living wife).
It is quite clear to any reasonable interpreter that polygyny was fully legal under the Mosaic Covenant. This is an embarrassing fact to a lot of Christians, because most of them teach (correctly, in my view) that Jesus made polygyny equivalent to adultery (thus illegal) in the New Covenant (see Matt. 19:9).
This doesn't mean that polygyny was wrong or evil under the Old Covenant. Polygyny actually served two very good purposes:
First, God had ordered His people into various battles, which resulted in the deaths of many men. These deaths would inevitably cause at least an entire generation in which the normal 50/50 balance of women to men was skewed. With a one-to-one pairing of husband to wife, there would be a large excess of unmarried women. In God's eyes, it is "not good" for either a man or a woman to be alone (Gen. 2:18), maritally (except in rare circumstances: Matt. 19:11-12). Therefore, polygyny provided an opportunity for these women to be in a marriage covenant so that they would not be tempted to go to the Gentiles for husbands.
Second, God wanted the tribal inheritance lines to continue, at least while the Mosaic Covenant was in effect. He cared about this enough that He created the institution of Levirate marriage (Deut. 25:5-6), in which a childless widow could marry her deceased husband's brother in order to produce offspring to continue the family line. This was despite the normal prohibition of marrying a brother's wife (Lev.18:16), so it shows how important the issue of tribal inheritance was. Under these circumstance, when a wife was barren, it was completely acceptable (and legal) for a man to take another wife in order to provide offspring. There are multiple Biblical examples of this.
In no way was polygyny considered an "optimal" marriage arrangement, even in the Old Covenant. It was simply an acceptable (legal) response to sub-optimal circumstances (death in warfare and barrenness). We can speculate as to why Jesus closed off polygyny as an option: in the New Covenant, warfare will not be a norm, and tribal inheritance tied to the land is now irrelevant. Barrenness can be solved by adoption. Thus, we are back to the optimum: one man and one woman.
There must be factions among you... (1 Cor. 11:19)
If Jesus' prohibition of polygyny is the correct interpretation of Matt. 19:9 (and I believe it is), then it brings a major implication: polygyny must be punished by the civil government in the same way as adultery. Even if you think that the civil penalty for adultery (Lev. 20:10) has been abolished — I do not — then you still probably believe that unrepentant adultery is grounds for excommunication. Therefore polygyny must also be grounds for excommunication.
There are Christians who believe that polygyny is allowable under the New Covenant. I'm not talking about Mormons, who are followers of a false prophet; they are not Christians. I'm also not talking about Christians who use divorce to engage in "serial polygyny" (although there are lots of those around). I'm talking about Christians like Martin Luther:
Martin Luther personally thought polygyny was a bad idea, but he had no Biblical argument against it. I am not going to deny the label "Christian" to someone who honestly believes (like Martin Luther did) that polygyny is still OK. I disagree, but it is a genuinely debatable point. I would not even separate from a Christian brother who still supported polygyny verbally, but did not practice it. But if a Christian brother decided to practice polygyny, I would have to separate from him. If the act of polygyny becomes equivalent to adultery (as Jesus said in Matt. 19:9) then polygyny is both a crime and an excommunicable offense. It cannot be merely a "conscience issue," like observance of food laws, or feast days.
Therefore, differing interpretations can lead to separation, on issues which have civil government implications. A Christian community can either be "pro-polygyny" or "anti-polygyny" when it comes to law. This doesn't mean that one Christian community can impose its law by force on everyone else (the way that Christians imposed their anti-polygyny law on Mormons in 1878; see Reynolds v. United States). That is not what Jesus intended: Matt. 20:25. But we should draw a civil government boundary line between us. This difference in interpretation leads, logically, to two different governments (two different jurisdictions). God will eventually show which "faction" is genuinely following His will (1 Cor. 11:19).