There are right and wrong ways to deal with controversy and error:
This may not be a really difficult passage as such, but it can cause some confusion or questions as to how we apply it. It seems to offer somewhat conflicting advice, when in fact it simply offers differing but related instructions to similar sorts of issues. The passage says this:
Don’t have anything to do with foolish and stupid arguments, because you know they produce quarrels. And the Lord’s servant must not be quarrelsome but must be kind to everyone, able to teach, not resentful. Opponents must be gently instructed, in the hope that God will grant them repentance leading them to a knowledge of the truth, and that they will come to their senses and escape from the trap of the devil, who has taken them captive to do his will.
 Verse 23 speaks of not getting into arguments; verse 24 urges us not to quarrel; then verses 25-26 urge us to correct others. A superficial reading of this text of Paul to Timothy may seem to present discordant instructions, but not really. What does seem hard to reconcile is the famous pair of verses found in Proverbs 26:4-5:
Do not answer a fool according to his folly,
    or you yourself will be just like him.
Answer a fool according to his folly,
    or he will be wise in his own eyes.
Now that does seem to be contradictory! But even there, it is not as such. See my earlier piece offering a resolution on this here: https://billmuehlenberg.com/2018/12/27/difficult-bible-passages-proverbs-264-5/
Here we have differing aspects to the issue of how best to respond to disagreements, arguments and error. The various responses form a coherent whole. Yes, those who just like to argue and who get their kicks out of picking all sorts of intellectual or theological fights are to be avoided. There is no need to go down that path with these folks.
And it is also true that the man of God must not be quarrelsome. But that is NOT to say we should avoid all controversial matters, or wink at obvious false teaching. The final two verses make this clear: we ARE to take head on those who oppose the gospel message and seek to distort it. Those sorts of fights we are not to run away from.
As such we have something we find often in the epistles of Paul: a call to both orthopraxis and orthodoxy. The former has to do with right living, while the latter has to do with right teaching. Both are vitally important, and both must be championed side by side.
Paul makes this clear in places like 1 Timothy 4:16 where we find these words: “Watch your life and doctrine closely. Persevere in them, because if you do, you will save both yourself and your hearers.” So it is never a matter of choosing one over the other, but of affirming both simultaneously.
Thus Paul tells Timothy that he should not quarrel or get entangled in foolish and pointless arguments (this is part of guarding or watching your life). And he tells him to stand for sound teaching over against his opponents (this is part of guarding or watching your doctrine).
It is about getting the biblical balance right, in other words. Refusing to be drawn into wasteful and useless arguments does NOT mean we just allow false teaching to run riot. We can stand up for biblical truth while also not getting involved with those who just like to debate stuff and argue all day.
Let me offer some commentary from others on these matters. Robert Yarbrough says this:
Admittedly there is a tension in this matter. At the same time Timothy should stay out of certain kinds of conflicts in some and perhaps most situations, he should “fight the good fight of faith” (1 Tim 6:12), publicly rebuke elders who are sinning (1 Tim 5:20), guard the good deposit (2 Tim 1:14), suffer like a military combatant (2 Tim 2:3), and perform other proactive if not aggressive functions. No pastor can or should always avoid disputes; to do so is to ensure that wolves take over the flock (see Acts 20:28-30). The Gospels support an interpretation of Jesus as a sinless man, but they are at the same time a study of conflicts he engaged in. The key in 2 Tim 2:23 may be “foolish and stupid.” Some arguments are of the no-win variety. It takes pastoral wisdom steered by divine guidance, whether through Scripture or Spirit or human advisors, to recognize them when they arise and to act (or not) accordingly.
The ever wise and theologically astute Gordon Fee said this about this text: “Not to quarrel does not mean that he must thereby let error go on its way. To the contrary. However, in standing against error he must exhibit a different disposition. He must be kind to everyone—even to his opponents. The point is attitudinal and reflects the very difficult stance of Ephesians 4:15 (‘speaking the truth in love’).”