STARTING LINE
Replacement Theology has had a wide-ranging impact upon the Church by veiling the overview of God's ongoing purposes through Israel.
The impact of Replacement Theology comes from how it excludes any ongoing purpose of God through Israel, erasing the vision of God finally revealing His glory through Israel, and therefore eroding the expression of glory "in the Church throughout all generations" (Ezekiel 8:1-4 & Ephesians 3:21). Which is very serious given this could be its last generation.
"Now to Him who by the power at work within us is able to do far more abundantly than all we ask or think, to Him be glory in the Church and in Christ Jesus to all generations forever. Amen." (Ephesians 3:20)
Much of the Church (at least in the West) believes it's directly acquired the glory God first placed upon Israel. Leading, if only by default, to the ignorance and arrogance Paul warned of in his olive tree analogy where the branches proudly believe they support the roots (Romans 11:18).
As illustrated by how it resorts to religious practices to fill the gaps left by its disconnection from its Jewish roots, to then present them as true expressions of Christianity when they're actually man-made.
"I hate, I despise your feasts, and I take no delight in your solemn assemblies. Even though you offer Me your burnt offerings, I will not accept them ... Take away from Me the noise of your songs: to the melody of your harps I will not listen. But let justice roll down like waters, and righteousness like an ever-flowing stream." (Amos 5:21-24)
Less obvious, but even more consequential, is the Church's blindness to Israel still being "the apple of God's eye" (Zechariah 2:8). So that it misses the depth of God's love for His chosen people and makea itself vulnerable to the curse assigned to those who don't bless Abraham's descendants (Genesis 12:3). Which can lead to anti-Semitism that, in the extreme, releases the type of hatred seen in the Spanish Inquisition and the Holocaust.
Furthermore, its gospel gravitates towards 'what this means for me' and 'how I can get saved and go to heaven', in great contrast to the first believers' gospel that empowered them to glorify the God of Israel by being "a light to the nations" (Isaiah 49:6).
All of which has made the Church like a skyscraper built across a fault line, with one side based on the truth of "salvation by grace through faith" (Ephesians 2:8) and the other on the lies of Replacement Theology, that wobbles when the two sides move in opposite directions, and prompts those inside who don't have God's reference point of Israel to reach for their particular theological grab-handle.
"I pray also for those who believe in Me through their word, that they may all be one." (John 17:20-21)
Replacement Theology's impact then escalates from denying the Church what it should have, to enable warped beliefs or a disproportionate emphasis on one particular belief to enter in. To fuel its split into 35,000 denominations in contradiction to Jesus' prayer for unity in John 17, that further reduces its ability to give glory back to God.
"Justice is far from us, and righteousness does not reach us. We look for light but all is darkness; for brightness but we walk in deep shadows. Like the blind we grope along the wall, feeling our way like men without eyes ... " The Redeemer will come to Zion, to those in Jacob who repent of their sins, declares the Lord." (Isaiah 59:9-10 & 20)
So, just as Paul answered his question "what advantage has the Jew?" by saying "much in every way" (Romans 3:1-2), the same can be said about the disadvantage of believers who are under the veil of Replacement Theology.
FINISHING LINE
The Church has been greatly disadvantaged by believing it has replaced Israel and not recognising what it's been included into.