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How do different Protestant denominations view the relationship between Israel and the Church?
The different views frequently track what a denomination teaches about end times.
There can also be various views within a denomination. The views run a spectrum.
There is no more Israel (as far as God is concerned) and the church has inherited all of the promises made to Israel. This is frequently called replacement theology.
God is finished with Israel and now and in the future will only deal with humanity through the church. This view is often denied as being the same as replacement theology, but it does sound similar.
Israel is irrelevant and God may or may not do something with humanity in the future. This might be called de-facto amillennialism. Here there may not be any specific teaching or doctrine just more of a focus on the present.
A more formal view of amillennialism is that at some future point God will return to earth and either make the world better or receive a world improved by Christians. This view was more popular when it was parallel to the political progressivism of the early 20th century. WWI took a lot of the wind out of its sails. This view is more popular with covenantal, reformed, reconstructionist, and dominionist theologies. This view has no regard for Israel other than just another nation from which some might be called to be Christian.
A millennial view would be one that sees Jesus returning to rule on earth for a thousand years. Dispensationalists often hold this view. This view is also often associated with a restoration of a faithful remnant of Israel such that all of the promises made to the nation of Israel will be fulfilled to Israel as promised.
There is a less popular perspective that can fall under the category of Messianic Judaism where Christians observe some of all of the laws given to Israel and may see no place for a separate gentile church.
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It sounds like you are wondering about the debate between covenant theology and dispensationalism. These are big questions, and only a very high-level summary can be provided here.
Covenant theology holds that there is one people of God, and that the saved from both Israel and the Christian church are members of that people. Thus, God's promises to Israel in the Old Testament apply to the Christian church as well.
Dispensationalism sees Israel and the Christian church as distinct groups, and that the promises to Israel in the Old Testament generally do not apply to the Christian church.
Take note too that there are many other categories besides these two frameworks. Ideas like dual-covenant theology and supersessionism ("replacement theology") are sometimes seen to be more extreme forms of dispensationalism and covenant theology, while frameworks like New Covenant Theology are seen to be somewhere in the middle.
Regarding denominations, remember that there are thousands of Protestant denominations, so the best we can do on a site like this is a high-level summary. That said, Reformed and Presbyterian denominations tend to be adherents of covenant theology. Baptist denominations and especially "non-denominational" churches tend to be dispensational. But most denominations do not take an official stance, or their stance is general enough that it allows for friendly disagreement within their churches.