The Arkenstone, a.k.a. The Heart of the Mountain was buried at the end of The Hobbit. Was it recovered at some later date?
After Thorin was killed, it was buried with him in his tomb.
It seems unlikely that something that valuable wouldn't have been dug up at some later date by goblins or orcs, or some other creature.
The Hobbit and The Lord of the Rings take the chronology of Tolkien's Middle-earth only as far as a couple of centuries after the end of The Lord of the Rings. This is the time which the Wise often call "The Dominion of Men":
The Elder Days are gone. The Middle Days are passing. The Younger Days are beginning. The time of the Elves is over, but our time is at hand: the world of Men, which we must rule.
(Saruman, in The Lord of the Rings, Book II, Chapter 2, "The Council of Elrond")
One can guess, or could guess, that at some point during this time Erebor would be again deserted by the Dwarves, and the Arkenstone—if it were not taken by the last to leave the Mountain—would be left to be looted by later "explorers". But we are not told this in any of the published writings. The last we are told about the dwarves of Erebor is that
After the fall of Sauron, Gimli brought south a part of the Dwarf-folk of Erebor, and he became Lord of the Glittering Caves.
It thus appears that the kingdom of Erebor persisted for at least a while after the end of the Third Age; the copy we have of the Red Book is that which was edited by Findegil, scribe of Gondor, in 175 of the Fourth Age, and it apparently contains no reference to the disappearance or decline of Erebor. There is no reference to what might have happened after that.
Tolkien doesn't say.
His latest writing on the fate of the Arkenstone is in Appendix A of Lord of the Rings, and that just amounts to a single sentence:
Thorin Oakenshield was mortally wounded; and he died and was laid in a tomb under the Mountain with the Arkenstone upon his breast.
However, following the Battle of Five Armies, we do know that the Dwarves (under Dain Ironfoot) reclaimed Erebor and set up a new kingship there, so any future bandits/etc would of course have a Dwarf kingdom to get through if they fancied some grave-robbing.
As to its eventual fate in the Fourth Age and beyond, after the dwindling of the Dwarves, Tolkien just didn't write those stories. In the end we just don't know.
Presumably the tomb is located somewhere inside the Lonely Mountain. In that case it would be well protected from grave robbers, since
after the battle of Five Armies the mountain was once again a dwarvish city and fortress, inhabited by dwarves who migrated from the Iron Hills.
(I can't recall if there is any confirmation in the book that this is the case, but it would make sense.)
When Bilbo Baggins found it on Smaug's golden bed deep inside the Lonely Mountain, he pocketed it, having learned how much Thorin valued it. While the Dwarves with Thorin sorted the treasure, Thorin sought only the Arkenstone, unaware that Bilbo was hiding it in his pillow.[3] When the Dwarves refused to share any of the treasure with King Thranduil and Bard, who had killed Smaug the Magnificent, Bilbo crept out of the Dwarves' fort inside the Mountain, and gave them the Arkenstone; Bard, Thranduil, and Gandalf then tried to trade it for Bilbo's fourteenth share of Smaug's hoard. An evil army arriving from the Grey Mountains interrupted the dispute, the Battle of the Five Armies ensued, and Thorin was killed. The Arkenstone was placed upon Thorin's chest within his tomb deep under the Lonely Mountain, and so was returned to the earth at last.
