(Rom 11:16a) If the part of the dough offered as firstfruits is holy, then the whole batch is holy;
Paul sees the Gentiles and Jews as part of the same whole. The faithful Jews are the firstfruits sacrificed at the harvest in celebration. Their holiness makes the rest of the harvest holy.
(Rom 11:16b-) if the root is holy, so are the branches. 17 If some of the branches have been broken off, and you, though a wild olive shoot, have been grafted in among the others and now share in the nourishing sap from the olive root, 18 do not boast over those branches. If you do, consider this: You do not support the root, but the root supports you.
Israel is the root — holy and supporting the wild branches — the Gentiles — grafted in. Remember, Paul says, do not despise the Jews. Your salvation comes from the root they provide.
Paul’s image is not original.
(Jer 11:16-17) The LORD called you a thriving olive tree with fruit beautiful in form. But with the roar of a mighty storm he will set it on fire, and its branches will be broken. 17 The LORD Almighty, who planted you, has decreed disaster for you, because the house of Israel and the house of Judah have done evil and provoked me to anger by burning incense to Baal.
Once again, Paul is indirectly — but pointedly — comparing the Jews who reject Jesus to the idolaters in Judea just before the destruction of Solomon’s temple by Nebuchadnezzar. This is what causes the natural branches to be broken off the tree!
(Rom 11:19-20a) You will say then, “Branches were broken off so that I could be grafted in.” 20 Granted. But they were broken off because of unbelief, and you stand by faith.
Very plainly, Paul explains that Israel’s branches were broken off because of unbelief (in parallel to idolatry). Unbelief results in damnation. Faith results in salvation. Failure to believe in Jesus, who is God in the flesh, is the equivalent of idolatry. It’s a rejection of God himself, and so the results will be much the same as before.
(Rom 11:20b-22) Do not be arrogant, but be afraid. 21 For if God did not spare the natural branches, he will not spare you either. 22 Consider therefore the kindness and sternness of God: sternness to those who fell, but kindness to you, provided that you continue in his kindness. Otherwise, you also will be cut off.
Paul then warns the Gentiles: your salvation is not guaranteed. Your branches can be broken off, too! Remaining attached to the life-giving root is conditional.
(Rom 11:23-24) And if they do not persist in unbelief, they will be grafted in, for God is able to graft them in again. 24 After all, if you were cut out of an olive tree that is wild by nature, and contrary to nature were grafted into a cultivated olive tree, how much more readily will these, the natural branches, be grafted into their own olive tree!
Israel can be restored to the root as well — if they come to faith in Jesus! God will be delighted to do just this. And Paul is hoping that his missionary work will bring this result.