Replacement theology is the belief that the Church has replaced Israel in God's covenantal plan. The Jews, allegedly, have been cast off, and the Church now inherits the blessings that were promised to them, albeit in a spiritualised sense.
No matter where you look in the Bible, you will never find such a doctrine. Quite the opposite, the Bible militates against this possibility. By far the best passage to show this is Romans 11.
This passage starts off rather explicitly by asking the question "has God cast away His people?" Paul's answer is "certainly not!" He then points out that he, himself, is a Jew, from the tribe of Benjamin. He then reiterates that God has not rejected His people, whom He foreknew.
After this, he brings up Elijah. In 1 Kings 19, shortly after the famous showdown with the prophets of Baal, Elijah was terrified for his life. The prophets of Baal had been killed, having been thoroughly refuted by God's miraculous power, and so Jezebel threatened to kill him. So Elijah fled. Feeling very much alone, he believed he was the last faithful Jew, and they were threatening to kill him, too.
But God's response, to which Paul harkens, is that there was a remnant of the faithful. 7,000 Jews had not worshiped Baal. By God's grace, not Israel's works, these Jews had been reserved. Similarly, by God's grace, many Jews, both in Paul's day and in ours, are saved, in spite of the works of Israel.
Paul then goes on to show that those who were not saved were actually blinded by God Himself. The elect of Israel obtained salvation, but the rest have been judicially hardened.
Paul continues to explain the reason for this hardening. First, he reiterates "have they stumbled that they should fall? Certainly not!" The concept of a fall carries the implication of total destruction. The nation has been abandoned, wiped out by God. But that wasn't going to happen. We still have the Jews today. Israel didn't stumble over Christ so that God would destroy them. Rather, the reason they rejected the Gospel is so that the Gentiles may be saved, which should in turn provoke the Jews to jealousy, because as gentiles, we have obtained what they sought. Then in verse 12, Paul asks "Now if their fall is riches for the world, and their failure riches for the Gentiles, how much more their fullness!"
This isn't just a throwaway line, either. Continuing with the concept of Jewish jealousy, Paul says "if by any means I may provoke to jealousy those who are my flesh and save some of them. For if their being cast away is the reconciling of the world, what will their acceptance be but life from the dead?" It's a truly interesting concept. Jews and Gentiles are effectively missionaries to each other. The Jews get the Gentiles saved by turning their hearts against God, but because the Gentiles get saved through this, the Jews become jealous, and so they turn to Christ, which according to Paul would be even more riches for the Gentiles.
Then we come to the famous olive tree analogy. This is truly devastating to Replacement Theology. As Gentiles, we are called "wild branches", being grafted to the root (the promise to Abraham). For this reason, we are told not to boast. The root sustains us, not vice versa. It is as Paul said in verse 6: If by grace, no longer of works. We didn't do anything to obtain these promises, neither did the "branches" that were broken off do anything to be broken off. Rather, they, them being the Jews, were broken off because of unbelief, whereas we were grafted on for our faith. Therefore, Paul tells us, in no uncertain terms, "Do not be haughty, but fear." Why? Because if God did not spare the "natural branches", He's not going to spare us, either.
Paul then highlights two very important aspects of God: His goodness and His severity. Those who were broken off receive His severity, but we receive His goodness. If we continue in His goodness. If not, He can cut us off just as easily.
Furthermore, Paul says that even those who were cut off can be grafted back in again. It's all about faith. You don't believe, you get cut off. You start to believe again, God can graft you back in. If we, as Gentiles, can be grafted in "against nature", how much more can Israel be grafted in, given that it's their tree? This is why it matters that Paul describes us as "wild" branches. This sort of thing is just bad farming. You don't graft wild branches onto cultivated plants. It won't produce the right fruit. But if a branch was taken from a tree in the first place, it can easily be put back.
With all that said, Paul tells us not to be ignorant of the fact that "blindness in part has happened to Israel until the fullness of the Gentiles has come in." Until. In other words, this isn't a permanent thing. There will come a time, when all the elect Gentiles believe, that the blindness of Israel will be cured. Then Paul quotes Isaiah 59:20-21 as a support for this.
Verses 28-29 eliminate the scorched ashes of Replacement Theology, telling us "Concerning the gospel they are enemies for your sake, but concerning the election they are beloved for the sake of the fathers. For the gifts and the calling of God are irrevocable." The Jews, for now, are the enemies of the Gospel. Most of them reject Jesus, and even set out to oppose it. But "for the sake of the fathers", they are beloved. God still loves the Jews, God still protects the Jews, God still continues to fulfill His promises to the Jews, because He made those promises to Abraham, Isaac, and Jacob. The gifts and the calling of God are irrevocable. It is literally impossible, even for God, to cast off the Jews, simply because He has promised not to, and He cannot deny Himself (2 Timothy 2:13).
Paul concludes his point by showing that just as we were once disobedient, yet have obtained mercy through their disobedience, the mercy we obtain makes mercy available for all. The rest of the chapter is a hymn of praise.
We see, then, that Replacement Theology is just plain wrong. The Church hasn't replaced Israel, and to say otherwise puts us in a very precarious position. It tells us that God lied to the Jews, and thus we could not trust His promises to us either. It tells us that God's promises can fail, so why would we trust Him? Far better to believe that everything in the Bible is true, which would mean God is far from done with His chosen people. Rather, through the Jews, the Lord will spread His mercy across the globe.