What role does Israel play in the Millennium?
In the Millennium, the surviving remnant of Israel will be a converted, natural (not born again) people who inhabit the land, rebuild the ruined cities, farm the ground, and serve the Lord through purified priestly worship and offerings at Jerusalem, while the sons of God reign over them and over the whole earth.
Scripture teaches that Israel's conversion happens suddenly, as it were in one day. Zechariah pictures the Lord removing the iniquity of the land in a single day, and Micah says the Lord will subdue their iniquities and cast their sins into the depths of the sea. Because of this cleansing, the Jewish remnant who survive the tribulation and the wrath of God are able to stand before the Lord and serve Him in a way they could not while still in their sins. This remnant is distinct from the Church; they are the natural seed of Abraham, not people who have been born again, and they remain mortal, subject to aging and death during the thousand years.
Once cleansed, Israel's role centers on rebuilding, farming, and worship. The remnant will build houses and live in them, plant vineyards and eat their fruit, and this time no enemy will take what they have labored for; as it is written, their days will be like the days of a tree, and they will long enjoy the work of their hands. The waste cities will be rebuilt, the desolate land tilled until it becomes like the garden of Eden, and the surrounding nations will recognize that the Lord Himself restored and replanted what had been ruined.
Israel also carries a priestly and sacrificial function in this age. On God's holy mountain, the whole house of Israel will serve Him there, and He will require their offerings and the firstfruits of their oblations. The sons of Levi will be purified like gold and silver so that the offering of Judah and Jerusalem may again be pleasant to the Lord as in former days. Everyday items, even the bells on the horses and the pots used in Jerusalem and Judah, will be marked "Holiness Unto The Lord," and no unclean thing (called here "the Canaanite") will be found in the house of the Lord anymore.
Jerusalem also becomes the center where the nations of the earth are required to come up yearly to worship the King and keep the feast of tabernacles, with drought and plague warned against any nation, such as Egypt, that refuses to come. So Israel's remnant, once purged, becomes the earthly nation dwelling securely in its own restored land, farming and building for lasting enjoyment, and standing as the center of worship and sacrifice under the reign of Christ and the glorified sons of God who rule over the whole earth.
The scriptures
Hear now, O Joshua the high priest, thou, and thy fellows that sit before thee: for they are men wondered at: for, behold, I will bring forth my servant the BRANCH. For behold the stone that I have laid before Joshua; upon one stone shall be seven eyes: behold, I will engrave the graving thereof, saith the LORD of hosts, and I will remove the iniquity of that land in one day. In that day, saith the LORD of hosts, shall ye call every man his neighbour under the vine and under the fig tree.
Who is a God like unto thee, that pardoneth iniquity, and passeth by the transgression of the remnant of his heritage? he retaineth not his anger for ever, because he delighteth in mercy. He will turn again, he will have compassion upon us; he will subdue our iniquities; and thou wilt cast all their sins into the depths of the sea.
And they shall build houses, and inhabit them; and they shall plant vineyards, and eat the fruit of them. They shall not build, and another inhabit; they shall not plant, and another eat: for as the days of a tree are the days of my people, and mine elect shall long enjoy the work of their hands. They shall not labour in vain, nor bring forth for trouble; for they are the seed of the blessed of the LORD, and their offspring with them.
For in mine holy mountain, in the mountain of the height of Israel, saith the Lord GOD, there shall all the house of Israel, all of them in the land, serve me: there will I accept them, and there will I require your offerings, and the firstfruits of your oblations, with all your holy things.
And he shall sit as a refiner and purifier of silver: and he shall purify the sons of Levi, and purge them as gold and silver, that they may offer unto the LORD an offering in righteousness. Then shall the offering of Judah and Jerusalem be pleasant unto the LORD, as in the days of old, and as in former years.
And I will bring again the captivity of my people of Israel, and they shall build the waste cities, and inhabit them; and they shall plant vineyards, and drink the wine thereof; they shall also make gardens, and eat the fruit of them.
Thus saith the Lord GOD; In the day that I shall have cleansed you from all your iniquities I will also cause you to dwell in the cities, and the wastes shall be builded. And the desolate land shall be tilled, whereas it lay desolate in the sight of all that passed by. And they shall say, This land that was desolate is become like the garden of Eden; and the waste and desolate and ruined cities are become fenced, and are inhabited. Then the heathen that are left round about you shall know that I the LORD build the ruined places, and plant that that was desolate: I the LORD have spoken it, and I will do it.
And it shall come to pass, that every one that is left of all the nations which came against Jerusalem shall even go up from year to year to worship the King, the LORD of hosts, and to keep the feast of tabernacles. And it shall be, that whoso will not come up of all the families of the earth unto Jerusalem to worship the King, the LORD of hosts, even upon them shall be no rain. And if the family of Egypt go not up, and come not, that have no rain; there shall be the plague, wherewith the LORD will smite the heathen that come not up to keep the feast of tabernacles. This shall be the punishment of Egypt, and the punishment of all nations that come not up to keep the feast of tabernacles. In that day shall there be upon the bells of the horses, HOLINESS UNTO THE LORD; and the pots in the LORD’s house shall be like the bowls before the altar. Yea, every pot in Jerusalem and in Judah shall be holiness unto the LORD of hosts: and all they that sacrifice shall come and take of them, and seethe therein: and in that day there shall be no more the Canaanite in the house of the LORD of hosts.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.