What happens to death and hell at the end?
At the end, death and hell will be emptied of the dead for judgment, and then death and hell themselves will be cast into the lake of fire, which is called the second death.
Scripture teaches that Jesus, through His resurrection, holds the keys of hell and death. At the Great White Throne Judgment, He will use those keys to open death and hell and bring forth the wicked dead who have been held there, so that they can stand before God to be judged according to what is written in the books, including the book of life.
After this final resurrection and judgment take place, death and hell themselves are done away with. Revelation 20:14 says plainly, "And death and hell were cast into the lake of fire. This is the second death." So death and hell are not eternal powers that continue on forever; they are themselves judged and thrown into the lake of fire along with those who are found written in condemnation.
This happens at the close of a larger sequence of end time events. Satan is loosed from the bottomless pit, gathers the nations for a final battle, is defeated by fire from heaven, and is cast into the lake of fire. Then comes the Great White Throne Judgment, where the dead, small and great, stand before God and are judged out of the books that are opened. It is in this same season that death and hell, having served their purpose of holding the wicked dead until judgment, are cast into the lake of fire and destroyed as the second death.
What follows is a new heaven and a new earth, with the first heaven and first earth having passed away. The New Jerusalem comes down from God, and God dwells with His people forever. So the ending of death and hell clears the way for eternity, when God's people live with Him without the presence of death, darkness, or the enemy ever again.
The scriptures
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 when the thousand years are expired, Satan shall be loosed out of his prison, And shall go out to deceive the nations which are in the four quarters of the earth, Gog and Magog, to gather them together to battle: the number of whom is as the sand of the sea. And they went up on the breadth of the earth, and compassed the camp of the saints about, and the beloved city: and fire came down from God out of heaven, and devoured them. And the devil that deceived them was cast into the lake of fire and brimstone, where the beast and the false prophet are, and shall be tormented day and night for ever and ever. 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 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.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.