Will there be death during the Millennium?
Yes, death will still occur during the Millennium, though life spans will be greatly extended; people will die, but a child will live to be a hundred years old, and only a sinner dying at that age will be considered accursed.
Scripture teaches that death continues to operate during the Millennium, even though it is a time of righteousness and peace under Christ's reign. Isaiah 65:20 says there shall be no more an infant of days, nor an old man that hath not filled his days, for the child shall die a hundred years old, but the sinner being a hundred years old shall be accursed. This shows that death still happens, but it happens after a very long life, unlike anything we see today.
It is pointed out that death still reigning during the Millennium is notable because, going all the way back to the days of Adam, including Methuselah who lived nine hundred sixty nine years, no man has ever lived past one biblical day, meaning a full thousand years. So while lifespans in the Millennium will be greatly lengthened compared to our current experience, death has not been fully abolished yet during this thousand year reign.
This stands in contrast to eternity, when there will be no more death at all. Revelation 21:4 declares that 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. That promise belongs to the eternal state that follows the Millennium, not to the Millennium itself.
So the picture given is one of two distinct periods: during the thousand year reign of Christ, death still exists, though held back to an extraordinary degree so that dying young, even at a hundred years old, is treated as a curse reserved for the sinner. It is only after the Millennium ends, when eternity begins with the new heaven, new earth, and New Jerusalem, that death is finally done away with completely.
The scriptures
And I will rejoice in Jerusalem, and joy in my people: and the voice of weeping shall be no more heard in her, nor the voice of crying. There shall be no more thence an infant of days, nor an old man that hath not filled his days: for the child shall die an hundred years old; but the sinner being an hundred years old shall be accursed.
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 John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband. 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 the city had no need of the sun, neither of the moon, to shine in it: for the glory of God did lighten it, and the Lamb is the light thereof.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.