The following isn't it's final form although carries the main bulk of what I'd like to say, I will edit it in the next few hours as I find time.
Tolkien addresses the death of Ents in his 1944 letter to his son.
So ends the Middle Age and the Dominion of Men begins, and Aragorn far away on the throne of Gondor labours to bring some order and to preserve some memory of old among the welter of men that Sauron has poured into the West. But Elrond has gone, and all the High Elves. What happens to the Ents I don't yet know. It will probably work out very differently from this plan when it really gets written, as the thing seems to write itself once I get going, as if the truth comes out then, only imperfectly glimpsed in the preliminary sketch. ....
Although he states he will later, he never goes on to discuss the death of Ents. Some users on the internet claim that the Ents are Maia due to Letter 247 to Colnel Worskett
“No one knew whence they (Ents) came or first appeared. The High Elves said that the Valar did not mention them in the 'Music'. But some (Galadriel) were [of the] opinion that when Yavanna discovered the mercy of Eru to Aulë in the matter of the Dwarves, she besought Eru (through Manwë) asking him to give life to things made of living things not stone, and that the Ents were either souls sent to inhabit trees, or else that slowly took the likeness of trees owing to their inborn love of trees. (Not all were good [words illegible]) The Ents thus had mastery over stone. The males were devoted to Oromë, but the Wives to Yavanna.”
The argument being that since they're spirits sent by Eru, they must be Maia. This would mean they are bound to the same fate as the Maia, which is covered here.
I however disagree, from Treebeard's chapter in The Two Towers he discusses how as Ents age they become more Tree-like. Treebeard says of himself
And I do not sit down. I am not very, hm, bendable.
And later says of Quickbeam
He could bend and sway like a slender tree in the wind.
Treebeard further seems to suggest that a few simply become more "tree-ish"
Some of us are still true Ents, and lively enough in our fashion, but many are growing sleepy, going tree-ish, as you might say.
As a final note. One of the final things we get on death and Ents is the following passage. However this seems to be about the natural death rather than the ones who (may have) died in the fires in the Battle of Isengard
Why are there so few, when you have lived in this country so long?’ asked Pippin. ‘Have a great many died?’
‘Oh, no!’ said Treebeard. ‘None have died from inside, as you might say. Some have fallen in the evil chances of the long years, of course; and more have grown tree-ish. But there were never many of us and we have not increased.