When is the dragon cast into the lake of fire?
The dragon, Satan, is cast into the lake of fire at the end of the thousand years, after he is loosed from his prison and gathers Gog and Magog to battle against the beloved city; fire comes down from God and devours them, and the devil who deceived them is then thrown into the lake of fire and brimstone.
Scripture shows that Satan is bound for a thousand years and shut up in the bottomless pit so that he cannot deceive the nations during that time. When the thousand years are finished, he is loosed out of his prison for a little season.
Once released, he goes out to deceive the nations in the four quarters of the earth, called Gog and Magog, gathering them together as an army as numerous as the sand of the sea. They come up against the beloved city and surround the camp of the saints, but fire comes down from God out of heaven and devours them.
It is at this point, after that final battle and the fire from heaven, that the devil who deceived them is cast into the lake of fire and brimstone, joining the beast and the false prophet who are already there. There he is tormented day and night for ever and ever. This happens before the great white throne judgment, where the dead are judged and death and hell themselves are also cast into the same lake of fire.
So the casting of the dragon into the lake of fire comes after the millennium, following his final deception of the nations and their destruction by fire from heaven, and it precedes the resurrection of the wicked dead and the final judgment before the great white throne.
The scriptures
And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years, And cast him into the bottomless pit, and shut him up, and set a seal upon him, that he should deceive the nations no more, till the thousand years should be fulfilled: and after that he must be loosed a little season.
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 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.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.