What is the relationship between the church and Israel?

Summarising Wayne Grudem on this:

Dispensational teaching is represented by Lewis Sperry Chafer: God has two distinct plans for Israel and the church:

‘God’s purposes and promises for Israel are for earthly blessings, and they will yet be fulfilled on this earth at some time in the future. On the other hand, God’s purposes and promises for the church are for heavenly blessings, and those promises will be fulfilled in heaven. This distinction between the two different groups that God saves will especially be seen in the millennium, according to Chafer, for at that time Israel will reign on earth as God’s people and enjoy the fulfillment of Old Testament promises, but the church will already have been taken up into heaven at the time of Christ’s secret return for his saints (“the rapture”).’

According to progressive dispensationalists, such as Robert Saucy, Craig Blaising, and Darrell Bock, God has a single purpose, which is the establishment of the kingdom of God in which both Israel and the church share.  The OT prophecies concerning Israel are not fulfilled in the church, but will be literally fulfilled in the Millennium, which believing Jews who will live in the land of Israel.

Non-dispensationalists believe that the church encompasses OT and NT believers.  This view allows for a large-scale conversion of Jewish people in the end times, according to Rom. 11:12; cf. 11:15, 23–24, 25–26, 28–31.

‘Yet that conversion will only result in Jewish believers becoming part of the one true church of God—they will be “grafted back into their own olive tree” (Rom. 11:24). They will not constitute a different olive tree, a different group distinct from the church.’

Many NT passages teach that the church is the ‘new people of God’, the ‘new Israel’.

Hebrews 8

‘In the context of speaking about the new covenant to which Christians belong, the author of Hebrews gives an extensive quotation from Jeremiah 31:31–34, in which he says, “The days are coming, declares the Lord, when I will establish a new covenant with the house of Israel and with the house of Judah.… For this is the covenant that I will make with the house of Israel after those days, declares the Lord: I will put my laws into their minds, and write them on their hearts, and I will be their God, and they shall be my people” (Heb. 8:8–10). Here the author quotes the Lord’s promise that he will make a new covenant with the house of Israel and with the house of Judah, and says that that is the new covenant that has now been made with the church.’

James 1:1

He writes to ‘the twelve tribes in the Dispersion.

‘Evidently he views New Testament Christians as the successors to and fulfillment of the twelve tribes of Israel.’

1 Peter 1:1

Again, he writes to his mixed readership as ‘exiles of the Dispersion’ – a very ‘Jewish’ appellation.

‘Dispersion” (1 Peter 1:1) to the next-to-last verse in which he calls the city of Rome “Babylon” (1 Peter 5:13), Peter frequently speaks of New Testament Christians in terms of Old Testament imagery and promises given to the Jews. This theme comes to prominence in 1 Peter 2:4–10, where Peter says that God has bestowed on the church almost all the blessings promised to Israel in the Old Testament. The dwelling place of God is no longer the Jerusalem temple, for Christians are the new “temple” of God (v. 5). The priesthood able to offer acceptable sacrifices to God is no longer descended from Aaron, for Christians are now the true “royal priesthood” with access before God’s throne (vv. 4–5, 9). God’s chosen people are no longer said to be those physically descended from Abraham, for Christians are now the true “chosen race” (v. 9). The nation blessed by God is no longer said to be the nation of Israel, for Christians are now God’s true “holy nation” (v. 9). The people of Israel are no longer said to be the people of God, for Christians—both Jewish and gentile Christians—are now “God’s people” who have “received mercy” (v. 10).’

It seems clear then, that for Peter:

‘The church has now become the true Israel of God and will receive all the blessings promised to Israel in the Old Testament?’

Romans 2:28f

Paul insists that a person is a Jew who is one inwardly (by the Spirit), not outwardly (by the word).

‘Paul recognizes that though there is a literal or natural sense in which people who physically descended from Abraham are to be called Jews, there is also a deeper or spiritual sense in which a true “Jew” is one who is inwardly a believer and whose heart has been cleansed by God.’

Romans 4:11f

Abraham is the father not only of his physical descendants, but (in a deeper and truer sense) of those who walk in faith, as he did.

Romans 9:6-8

The true children of Abraham – constituting the true Israel – and not those born by physical descent, but those who have believed in Christ.

Romans 9:25

Paul quotes Hos. 2:23 to show that it is believers in Christ whom God considers to be ‘my people’.  This is also the teaching of Gal 3:29 and Phil 3:3.

Ephesians 2:12-20

It is believing Jews and believing Gentiles together who have been ‘brought near by the blood of Christ’.  They are ‘one body’, ‘fellow citizens’, and together constitute ‘the household of God’.

‘The entire passage speaks strongly of the unity of Jewish and gentile believers in one body in Christ and gives no indication of any distinctive plan for Jewish people ever to be saved apart from inclusion in the one body of Christ, the church. The church incorporates into itself all the true people of God, and almost all of the titles used of God’s people in the Old Testament are in one place or another applied to the church in the New Testament.’

Grudem, Systematic Theology, 2nd ed., ch. 44, sec. 5.