What is the biblical basis for the belief that some people will live eternally on earth?

Isaiah 35 is a chapter referring to the coming kingdom of God, complete with the blind seeing and the lame walking again. The beginning of the chapter describes this place as an earth-like location.

In addition, the Old Testament speaks a great deal about Zion and its place in Jerusalem. See a search for "Zion" in the Bible here. Prophetically, the Lord will dwell there among His people, and Zion appears on earth.

Zechariah 14 tells of the time that God will come to earth, and He will come to the Mount of Olives in Jerusalem. "The people shall dwell in it; and no longer shall there be utter destruction. But Jerusalem shall be safely inhabited" (14:11). Interestingly enough, when Jesus ascended to heaven in Acts, two men said that He will return in like manner (Acts 1:11)—and He had ascended from the Mount of Olives (Acts 1:12)!

These among many other verses speak of the coming kingdom of God, that it will be set on earth.

Hebrews 11:5 - By faith Enoch was translated that he should not see death; and was not found, because God had translated him: for before his translation he had this testimony, that he pleased God.

Paul and other apostles quoted from the Book of Enoch, which was read. That's not in most people's bibles, only considered scripture by Ethiopic church traditions.

Jude 1:14 - And Enoch also, the seventh from Adam, prophesied of these, saying, Behold, the Lord cometh with ten thousands of his saints,

Apparently Enoch also wrote down prophecies, so other than Qumran and Dead Sea Scrolls, the originals are not in the bible. There is the Slavonic records which can be considered the 2nd Book of Enoch. The Third Book of Enoch is in the Hebrew tradition after the 2nd temple period.

As for why or how resurrected beings can exist on the Earth, it's similar to the angelic tradition seen when Lott met angels and the angels ate food and could physically move him with their hand. Angels can manifest a physical form, as is true of the resurrected Jesus of Nazareth. You reconstruct the body you had, but without imperfections, and it is physical in the sense that a nailed hand still has a hole in it.

There are two types of resurrections in scripture. The Lazarus tradition and the human walking with god tradition. Enoch, Jesus, belong to the latter, while even the Apostles were said to have resurrected dead people, most likely by bringing the spirit back into the body and then healing the body. So in both traditions, it should be possible for resurrected individuals to live on Earth but that doesn't address why this is in certain lines or prophecies.

http://stronginfaith.org/article.php?page=114

A number of people in the bible are claimed to have arisen from the dead. But there is a different type of "Risen" that applies specifically to Jesus of Nazareth and to Enoch.

Without understanding the context of the signs in the heavens/sky, and the rest of revelation and certain parts of Old Testament tradition, it is hard to put the end times into context as a prophetic prediction. With a new heaven and a new earth, the laws of physics in this universe could be changed. The people might transcend and still exist, but the concept of what a city is or a nation is, would be operating by completely different rules. If people are living on earth, there's no reason it couldn't be the old earth. The new earth might have no conceptual relation to this one.

I would go with the idea that it isn't heaven or earth. People will get to choose or will be allotted inheritances based on what they can handle, as Paul once mentioned. Can't find that verse about the 3 kingdoms of heaven yet. Some people are not going to want to live in Heaven with all the restrictions of Divine Law. If the choice is that or the fire lake (which is found in 1 Enoch, not in the Old Testaments), it becomes a pretty doctrinal either or option. If the option is more free and diverse, then there is less of a problem. The people who want to live on a new earth can do so.

The Apocalypse speaks of a reign on the earth:

"Worthy are you to take the scroll and to open its seals, for you were slain, and by your blood you ransomed people for God from every tribe and language and people and nation, and you have made them a kingdom and priests to our God, and they shall reign on the earth." (Revelation 5:9–10, emphasis added)

This should probably be understood as an eternal reign (Revelation 11:15, 22:5) and not just for a thousand years (20:6). However, this might be understood as the new earth rather than the present one (21:1).

Jehovah's Witnesses draw the belief of living forever on Earth for a few reasons.

The first of which was God's original purpose for mankind when he placed Adam and Eve in the Garden of Eden. While this original purpose has been postponed by sin and Satan's challenge to God's sovereignty, it is still God's will for mankind.

Further, God blessed them, and God said to them: “Be fruitful and become many, fill the earth and subdue it, and have in subjection the fish of the sea and the flying creatures of the heavens and every living creature that is moving on the earth.”
—Genesis 1:28

For this is what Jehovah says,
The Creator of the heavens, the true God,
The One who formed the earth, its Maker who firmly established it,
Who did not create it simply for nothing, but formed it to be inhabited:
“I am Jehovah, and there is no one else.
—Isaiah 45:18

9 Remember the former things of long ago,
That I am God, and there is no other.
I am God, and there is no one like me.
10 From the beginning I foretell the outcome,
And from long ago the things that have not yet been done.
I say, ‘My decision will stand,
And I will do whatever I please.’
11 I am calling a bird of prey from the sunrise,
From a distant land the man to carry out my decision.
I have spoken, and I will bring it about.
I have purposed it, and I will also carry it out.
—Isaiah 46:9-11

Psalm 37 tells us of a time when the righteous ones will possess the earth forever.

9 For evil men will be done away with,
But those hoping in Jehovah will possess the earth.
10 Just a little while longer, and the wicked will be no more;
You will look at where they were,
And they will not be there.
11 But the meek will possess the earth,
And they will find exquisite delight in the abundance of peace.
—Psalm 37:9-11

The righteous will possess the earth,
And they will live forever on it.
—Psalm 37:29

Finally, Revelation 21 tells us that "death will be no more." Since death has never existed in heaven, this must refer to the earth.

3 With that I heard a loud voice from the throne say: “Look! The tent of God is with mankind, and he will reside with them, and they will be his people. And God himself will be with them. 4 And he will wipe out every tear from their eyes, and death will be no more, neither will mourning nor outcry nor pain be anymore. The former things have passed away.”
—Revelation 21:3, 4

(For a complete article, there's "What Is God's Purpose For the Earth?")