I think that this question is more complicated than it looks and several of the answers -- while thoughtful and well-researched -- may be trying to make things too simple and to deduce the best procrustean bed.
We know that
Morgoth himself the Valar thrust through the Door of Night beyond the Walls of the World, into the Timeless Void;
It seems clear that this "Timeless void" is the same as the Void in which Arda was englobed when Iluvatar created it:
Iluvatar arose in splendour, and he went forth from the fair regions that he had made for the Ainur; and the Ainur followed him.
But when they were come into the Void, Iluvatar said to them 'Behold your Music!' And he showed to them a vision, giving to them sight where before was only hearing; arid they saw a new World made visible before them, and it was globed amid the Void, and it was sustained therein, but was not of it. And as they looked and wondered this World began to unfold its history, and it seemed to them that it lived and grew.
Melkor had gone down into Arda at the beginning at the same time as the Valar. After his final defeat, the Valar thrust him out in such a way that he could not re-enter. How? Tolkien does not tell us, but we can make a reasonable guess.
Throughout the Silmarillion and LotR the invariable consequence of a creature doing evil is a diminishment of that creature's own being, it's own self. Since a creature's strength and power and ability to act in the world is in large part determined by its strength of being, the practice of evil diminishes the creature. (Melkor lost fighting power -- once the greatest of the Ainur he finally found the Elf Fingolfin to be a tough opponent -- and eventually lost his ability to change his body or even to heal his own wounds.) It is likely that by the end Morgoth simply lacked the capability to re-enter Arda unbidden so whatever was left of him was stuck in the Void: Unable to re-enter Arda and unwilling to re-enter the Halls of Iluvatar.
Both Sauron and Saruman diminished themselves likewise:
[In Sauron's case, when the Ring went into the fire,] it seemed to them that, black against the pall of cloud, there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.
[When Wormtongue killed Saruman,] To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing.
This is the last we hear of either of them, but we also have Gandalf's prediction of what would happen to Sauron if the Ring is destroyed:
then he will fall; and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed.
Note that Gandalf -- surely an expert if anyone is -- sees Sauron (or what's left of him) remaining in Arda, but so diminished that he is of no consequence. This also fits Saruman's fate, as far as we know it: He dies, his stunted and diminished spirit rises (perceived by the Hobbits as smoke), and turn to the West from which a wind arises and disperses whatever is left into the air of Arda.
In neither case is there anything at all to suggest a thrusting out of Arda into the Void beyond.
The explanation, I think, is that Morgoth actually was thrust out of Arda into the Void beyond, but the remnants of Sauron and Saruman went into the void of nothingness. This is entirely consistant with Tolkien's religious understanding of evil: Evil does not exist as a thing, but only good, with the scale running from God on one end to nothingness on the other. Morgoth, Sauron and Saruman were created high in goodness and through their own actions, the latter two anyway, through their own choices made themselves less and less good to the point where, at their deaths, they had diminished themselves to close to nothing at all.
Morgoth's Void may be real (though never forget that we're dealing with Mannish legends derived from what Elves told them of what they remembered hearing from the Valar), but Sauron's void and Saruman's is the void of personal nothingness. And that is where Gandalf sees the Witch-King heading.