Of the three main wizards who appear in The Hobbit and the Lord of the Rings (excluding the 2 Blue Wizards, who are mentioned but don’t actually appear within either work) Radagast is probably the least well known. Saruman the White, who betrays the order of the maiar for his own selfish purposes by corrupting Isengard and building an army of orcs to try to take the ring for himself, and Gandalf the Gray (then white) who is almost the exact opposite of this and guides and protects all of the kingdoms of Middle Earth against Sauron, both play a fundamental part in deciding the future of the world.
Radagast’s part in the story, however, is far more minimal, and despite his character being extended and elaborated upon within Peter Jackson’s film adaptations of The Hobbit, he actually appears more in the Lord of the Rings books. All that is really told of him is that he is a loyal and trustworthy fellow, and that he favors the birds and the beasts of the world than the people and the politics, and only involves himself in affairs is specifically required to by Gandalf, who is his cousin and oldest ally. But what happens to him during the Fourth Age of Middle Earth, when Saruman has been killed by his own servant Wormtongue, and Gandalf has departed back to the Undying Lands?
As the maiar originally came from across the sea to Middle Earth in the first place, it would be easy to assume that Radagast would want to claim a place aboard the ships sailing into the west, and return to a haven away from war and greed and suffering. But this would be forgetting that all of the things he has come to know and love, the entirety of the life that has shaped his character is on Middle Earth.
Radagast is a particular expert of the flora and fauna that grow and thrive around Mirkwood, where he has his abode, and much of his magic has developed within the herb lore and healing of this place. This is the same with the animals who depend on him for protection and care, like Sebastian the hedgehog whom he brings back from the brink of death after Mirkwood is tainted by the evil of the Necromancer. The creatures who live in the woodland are dependent on his magic and his guidance, so it is doubtful that he would choose to leave them in favor of the Undying Lands.
There is also the possibility to consider that Radagast has technically neglected some of the duties he was given by the Valar when he was sent to Middle Earth in the first place, and therefore may not feel welcomed or accepted back in Valinor. The maiar were sent to guide and protect the peoples of Middle Earth, to shape the future of the world without overly influencing or stunting its development.
As Radagast has chosen a path of creatures over people, and is therefore not particularly involved in the War of the Ring against Sauron, which could destroy the world entirely, he has somewhat failed to achieve the mission he was sent to do in the first place. Unlike Gandalf, who died for the quest and came back as the white wizard, fought alongside the brave members of the fellowship in every battle, and stepped in on many occasions to save pivotal characters and prevent disastrous tragedies (such as the unbalanced Denethor burning Faramir alive), Radagast was only involved through sending the Giant Eagles to the rescue, rather than directly helping himself.
It is therefore likely that when the Fourth Age of Middle Earth begins, and the immortal beings fade out of the world, Radagast disappears off into the East somewhere, forging a path for himself and his animals far away from the world of Men. If Middle Earth really is a precursor to this world, and eventually went on to become Europe, then perhaps Radagast is deep in the Siberian plains, surrounded by the wildflowers of spring.
As Radagast is a very limited character in the original books, it is very hard to say for definite where he may have disappeared to, Tolkien never provided an ending to his story, but fans believe that he is still out there somewhere, disguised from human eyes like tree-herders, the ents, who have faded back into the forests and fallen back into deep slumber without the elves in the world to keep them awake and teach them to talk, and like the hobbits, who are possibly still in their smials somewhere underground, in a rural and uninhabited place, seldom seen and very weary of ‘Big People’.