What happens to the Antichrist and the False Prophet at the end?
After the battle of Armageddon, the Antichrist and the False Prophet are cast into the lake of fire, while their armies are left dead upon the earth.
When the war of Armageddon reaches its end, all of Jesus' enemies except death itself are dealt with at once. The Antichrist and the False Prophet, who had deceived the nations and made war on the saints, are cast into the lake of fire. Their armies, gathered against the Lord, are struck down and left lying dead upon the earth.
At this same time Satan himself is seized and bound with a great chain, then cast into the bottomless pit and sealed there for a thousand years so that he can no longer deceive the nations. This leaves the earth free of the Antichrist's rule and of Satan's active deception for the length of the Millennium.
Because the armies of Armageddon lie dead across the land, the early years that follow are spent cleansing the ground itself. The birds and beasts of the field are called to a great feast on the flesh and blood of the fallen, described as the sacrifice of the mighty upon the mountains of Israel, and the people of the cities burn the weapons of war for seven years as the land is purified from the corruption left behind.
The scriptures
And I saw an angel come down from heaven, having the key of the bottomless pit and a great chain in his hand. 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 they that dwell in the cities of Israel shall go forth, and shall set on fire and burn the weapons, both the shields and the bucklers, the bows and the arrows, and the handstaves, and the spears, and they shall burn them with fire seven years: So that they shall take no wood out of the field, neither cut down any out of the forests; for they shall burn the weapons with fire: and they shall spoil those that spoiled them, and rob those that robbed them, saith the Lord GOD. And it shall come to pass in that day, that I will give unto Gog a place there of graves in Israel, the valley of the passengers on the east of the sea: and it shall stop the noses of the passengers: and there shall they bury Gog and all his multitude: and they shall call it The valley of Hamongog. And seven months shall the house of Israel be burying of them, that they may cleanse the land. Yea, all the people of the land shall bury them; and it shall be to them a renown the day that I shall be glorified, saith the Lord GOD. And they shall sever out men of continual employment, passing through the land to bury with the passengers those that remain upon the face of the earth, to cleanse it: after the end of seven months shall they search. And the passengers that pass through the land, when any seeth a man’s bone, then shall he set up a sign by it, till the buriers have buried it in the valley of Hamongog. And also the name of the city shall be Hamonah. Thus shall they cleanse the land. And, thou son of man, thus saith the Lord GOD; Speak unto every feathered fowl, and to every beast of the field, Assemble yourselves, and come; gather yourselves on every side to my sacrifice that I do sacrifice for you, even a great sacrifice upon the mountains of Israel, that ye may eat flesh, and drink blood. Ye shall eat the flesh of the mighty, and drink the blood of the princes of the earth, of rams, of lambs, and of goats, of bullocks, all of them fatlings of Bashan. And ye shall eat fat till ye be full, and drink blood till ye be drunken, of my sacrifice which I have sacrificed for you. Thus ye shall be filled at my table with horses and chariots, with mighty men, and with all men of war, saith the Lord GOD.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.