In the book The Two Towers Treebeard tells Merry and Pippin the sad story of the Entwives, who were lost (literally lost - they didn't die); he says the Ents believe that they "may meet [the Entwives] again in a time to come" and find a new home together where they can all live in peace. However, he also says:
"But it is foreboded that that will only be when we both have lost all that we now have"
This loss will take the form of Sauron "withering" the woods of the Ents, just as Sauron once destroyed the gardens of the Entwives.
Then Treebeard sings an Elvish song about the Ents and Entwives, which ends with the lines
Together we [Ents and Entwives] will take the road that leads into the West,
And far away will find a land where both our hearts may rest.
Granted, fading into the West and traveling to a far away land there is a constant theme among the Elves, and this is an Elvish song, not an Entish one, but Treebeard says the Ents love it as well, which suggests that it reflects not only Elvish thought, but also Entish thought.
All of this is very familiar, and echoes the persistent thread woven throughout Tolkien's work - Arda and Middle-earth are changing; the ancient world of mythology is transforming into the world of history in which we live today; the old order is passing away; the time of the Elves is over, and the time of Men is beginning; magic will fade and science will supplant it; this final war against Sauron will be the last time that evil is gathered into a single, personified entity, and henceforth, the one ultimate evil will be replaced by many lesser, more insidious, subtler evils; and so on and so forth.
My question is this: When Treebeard and the Elvish song speak of the Ents and Entwives finding a new, permanent home in the West, are they referring to Aman? Will the Ents and Entwives eventually go to the Undying Lands? The only other option seems to be that Tolkien thought the Ents would either die out or linger in Arda forever; in the latter scenario, the Ents would presumably still be around today, though I would imagine that they all became what Treebeard referred to as "treeish", more or less immobile, and largely silent. So if the Ents don't go to the Undying Lands, did they die out, go somewhere else, or did they, in Tolkien's imagination, linger on to modern times, however treeish they may have become?
Edit: Tolkien Gateway's entry on Ents has this to say about the fate of the Ents; the first citation (marked 6) is to Robert Foster's Complete Guide to Middle-earth, and the second (marked 14) is to Return of the King, specifically the chapter "Many Partings":
The Ents remained in Fangorn where they probably dwindled in the following Ages, but Galadriel wished to Treebeard that they would meet again in Tasarinan, when Beleriand is lifted again from the waves, in Spring.