The “foolish questions” Paul is warning Titus about are questions about the Law. You know that because he mentions the law in this verse (v.9) and because of the context. Everyone knows we should maintain good works (v.8), but some in Crete were saying to get believers to do good works that you have to put them under the Law. So they constantly affirmed the Law! Paul says grace (v.7) should be constantly affirmed.
But when grace believers affirm grace, those who teach the Law always question it. We know their questions are “foolish” because that’s what Paul called the Galatians (twice!) who had gone back to the Law (Gal. 3:1-5). Fools are people who don’t obey God (I Sam. 13:13; Mt. 7:24-26; Gal. 3:1), and when you return to the law you’re not obeying the truth (3:1)
“Genealogies” (3:9) were part of the Law, so Paul said to avoid them too. They were needed in time past to determine who could serve God (Neh. 7:64), but you don’t have to be a Levite today to serve God, so genealogies are not needed!
Those who were teaching the Law were causing “contentions,” which should also be avoided (v.9). Those contentions were caused by pride (Pr. 13:10), Jewish pride in the Law (Rom. 2:17,18). This was a problem in Ephesus too (II Tim. 2:14), and the answer in both cities was “rightly dividing the word” (II Tim. 2:15) and realizing we’re not under the law (Rom. 6:15). So strivings about the law are “unprofitable” when it comes to maintaining good works (Tit. 3:9).
The Law used to profit the Jews who were under it, for God blessed them with health and wealth if they were good. Wouldn’t you maintain good works knowing you’d be rewarded like that? Of course, God is not doing that today, but if you think He is, the Law will become unprofitable to you because you’ll think God is letting you down! That’ll make you quit doing good works, or at least discourage you.
Paul says the Law is “vain,” which connects it to idolatry (I Sam. 12:21; Isa. 44:9 cf. Ps. 115:4,5; Jer. 16:19,20). Paul uses the word vain because the Jews had made a god out of the Law. If you think that can’t happen to something God gave, see II Kings 18:1-4.
People still worship the Law today, insisting we are under it, and they are stubborn about it. So Paul says to admonish them twice, then reject them (3:10). Heretics teach heresy, which is a doctrine contrary to established doctrine that has been accepted by God’s people. The Jews accused Paul of it, but he believed their established doctrine of the Law (Acts 24:14). He just taught that Gentiles weren’t under it! But for grace believers to return to the Law was heresy, and Paul says to reject heretics after two admonitions.
Admonitions are primarily the pastor’s job (I Thes. 5:12,13), but all grace believers know we’re not under the Law and so can admonish heretics (Rom. 15:14).
But remember, admonitions should be gentle (II Thes. 3:14, 15), and one of the gentlest ways we admonish one another is in the hymns we sing (Col. 3:16).
Heretics must be rejected after two admonitions because if they don’t show any inkling of being open to the truth, you are wasting the time you should be redeeming (Eph. 5:16).
“Subverted” (3:12) means to turn something upside down. They accused Paul of turning the world upside down (Acts 17:6), but he was not subverting anyone. And once his doctrine of grace was established, to teach contrary to what he taught was subversive! When Paul started preaching grace to Gentiles, some Jews didn’t like it and raised a fuss, so the apostles met with Paul at the Jerusalem council (Acts 15). They recognize Paul’s gospel and so wrote to the Gentiles to say they hadn’t authorized the “subverting” that those Jews were trying to cause in the Gentiles (Acts 15:24).
Paul says those who teach the Law to those under grace “sinneth” and condemn themselves with their words (3:11). They won’t admit that they condemn themselves with their words, but anyone who knows grace will know the Law when they hear it. Do you?