What generally happens is that the OT is used to support the direction and mission of the church in ways that cannot be sustained by the NT, which has more direct relevance to this question.
A favorite text to support this is Jeremiah 29. It is said that we, like Israel in exile, are to seek the welfare (peace, renewal, etc.) of our cities. Not that there is anything wrong, in themselves, of such ideas, but they should never be taught and practiced as the primary mission of the church. Israel was a theocratic society, whose role in culture and the world don't automatically translate to the church.
Bottom line: without understanding the biblical/theological purposes behind Israel's exile, there will be misunderstandings & misapplications galore. For example, the fact that they are there for 70 years is decided by God as the period of punishment/exile justified by their continual neglect of Sabbath years (2 Chr 36:20-21), and in just a few chapters later (Jeremiah 50-51) God speaks a harsh prophecy of judgment against Babylon.
Two of the primary questions that the Israelites were agonizing over during and after the exile was:
- what has happened to the promises to Abraham? God had promised land, descendants and relationship to Abraham and his descendants. Israel's ejection from the promised land seems to threaten these promises (esp. promise of land).
- Further, what has happened to the promises to David? God promised David a son to reign perpetually over Israel. The exile of David's descendants put these promises in jeopardy.
Also, Psalm 137 is written from the perspective of an exile who definitely does not pray for Babylon's peace, but rather prays for its destruction:
O daughter of Babylon, doomed to be destroyed,
blessed shall he be who repays you
with what you have done to us!
Blessed shall he be who takes your little ones
and dashes them against the rock! (v.8-9 ESV)
If our view of Israel's exile cannot reconcile the reality of faithful Israel's prayers for its peace and its destruction, then we have a faulty view. It was for Israel's sake that God said to "pray for the peace of the city," not for the cities' sake. It was to keep the promises to Abraham and David alive.
But, all these arguments to address the problem do it without bringing in one of the strongest arguments: the NT knows nothing of this kind of theology. Cities are useful hubs to spread the gospel to the surrounding areas, not God's "preferred method of culture building." That idea can only be supported with presuppositions from theocracy (specifically a city, Jerusalem, from which God rules his people).