In Tolkien's lore, they are many very old beings. Is there any indication that some or even one would have been truly immortal and survived into what we would call "our" history or Age? Galadriel, Cirdan, Ingwe, Bombadil, Ungoliant, or any Maia or Ainur? Anyone or anything? Seems there must be some kind of carry-over seeing as how "man" existed in this time.
Tolkien does better than hint. In the beginning of The Hobbit he states that hobbits may continue to exist now and that we don't notice because they keep out of our way. This is from the opening page:
He may well have gone back on this suggestion at some point, but it might be worth looking out for them next time you're in the English countryside. You'll have to be quiet though because:
All the immortal beings - elves, Ainu, etc - are still around.
Tolkien is quite clear that the Elves at least are "bound to the circles of the World" and will endure as long as it does. They may not be present in Middle-Earth itself, but they are certainly still alive somewhere.
The Vala and Maia also, when they descended into Arda, committed themselves to existing for as long as it does, and cannot leave before the end.
Eru Ilúvatar: He is The One, he was since ever and will be forever, but he is not "here" on Earth.
The Ainur (not Valar/Maiar): They are with Eru, also out of this world.
The Valar and the Maiar: They are eternal as well, and they are on Arda until the end of the world, but they are on Valinor, which is now removed from the world as we know it.
Elves: Those who were on Valinor continue there as the Valar and Maiar. Those who were on Middle Earth (either after return (Noldor) or those who never went to Valinor) left Middle Earth through the "straight path" towards Valinor. They'll be there, either live or dead on Mandos' "forever hotel", until the end of the world.
Dwarfs, hobbits: not immortals. Species may be still around, but no individuals. Ringbearers, however, are in Valinor.
Men: not immortals, by definition. Well, Duncan McLeod...
Orcs: Let's hope they were all killed.
Ents: It is unknown if they are truly immortal. If they are like trees, they are not, but they live way long. Some of the forests in Central Europe may have Treebeard still around.
Tom Bombadil: As we do not know who he really is, we can not know. But it is a good bet to say he is still in his small realm, probably around central France.
Note: Central Europe and Central France locations move a lot depending on the equivalences you do between northwestern Middle Earth and modern Europe.
EDIT: I forget about Arien and Tillion, who drive the Sun and the Moon, respectively. They are undoubtedly still performing their work.