What happens to the old heaven and earth?
The old heaven and earth pass away entirely, fleeing from the presence of God on His throne, and He replaces them with a new heaven and new earth where there is no more sea and no more death, sorrow, or pain.
When the great white throne judgment is set, the old heaven and earth flee away from the face of the One who sits on the throne, and no place is found for them. The earth and the heavens themselves come under fire; God purges the old creation because He is about to do a new thing.
After the wicked dead are raised and death and hell give up all that is in them, the old order is finished. Death and hell themselves are cast into the lake of fire, which is called the second death, along with anyone not found written in the book of life. The former heaven and earth, having passed away, are no longer needed for what God is about to bring forth.
In place of the old, the Father creates a new heaven and a new earth. There is no more sea in it, because God desires no separation and no division between nations or between the people of God. The tabernacle of God comes to dwell with men, and He wipes away every tear; there is no more death, sorrow, crying, or pain, for the former things have passed away.
This new earth becomes a paradise once again for mankind, even better than before. A pure river of the water of life flows from the throne of God and of the Lamb, with the tree of life bearing fruit for the healing of the nations, and mankind is restored to speaking one pure language so that all may call upon the name of the Lord together.
The scriptures
And I saw a great white throne, and him that sat on it, from whose face the earth and the heaven fled away; and there was found no place for them.
And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works. And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works. And death and hell were cast into the lake of fire. This is the second death. And whosoever was not found written in the book of life was cast into the lake of fire.
And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea.
And I heard a great voice out of heaven saying, Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God. And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away.
And he shewed me a pure river of water of life, clear as crystal, proceeding out of the throne of God and of the Lamb. In the midst of the street of it, and on either side of the river, was there the tree of life, which bare twelve manner of fruits, and yielded her fruit every month: and the leaves of the tree were for the healing of the nations.
For then will I turn to the people a pure language, that they may all call upon the name of the LORD, to serve him with one consent.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.