The immortality of Ents is a rather difficult topic. Like everything to do with Ents it is never really covered and assumptions have to be made from what we are told in the text.
The main argument against "immortality" seems to be this quote from Treebeard. I've repeatedly seen it used to described the "death" of Ents (by means other than damage to their body or soul, in the same way Elves can "die")
‘The trees and the Ents,’ said Treebeard. ‘I do not understand all that goes on myself, so I cannot explain it to you. 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. Most of the trees are just trees, of course; but many are half awake. Some are quite wide awake, and a few are, well, ah, well getting Entish. That is going on all the time.
The Two Towers, Book III, Chapter 4: Treebeard
This quote on the surface seems to suggest that Ents can at one point in their lifespan become sleepy and stop moving. Some trees however become quite lively and can be said to be becoming "Entish". Whether this means they've possessed the spirit of another Ent who's gone "tree-ish" or it is just an active tree described as being more Ent-like than regular trees is anyone's guess.
‘The New Age begins,’ said Gandalf, ‘and in this age it may well prove that the kingdoms of Men shall outlast you, Fangorn my friend.
Return of the King, Book VI, Chapter 6: Many Partings
Towards the end of the Return of the King, Gandalf again suggests that Ents may not be "immortal" saying that the kingdoms of Men may outlast even him.
As per my answer to a question about Treebeard's age, I think it's very probable that Ents are incredibly long lived and "immortal". Further, in the chapter "Many Partings" Galadriel suggest to Treebeard that they may meet again when the world is renewed (After Dagor Dagorath, the final battle) and will walk again in the spring.
‘It is long, long since we met by stock or by stone, A vanimar, vanima´lion nostari!’ he said. ‘It is sad that we should meet only thus at the ending. For the world is changing: I feel it in the water, I feel it in the earth, and I smell it in the air. I do not think we shall meet again.’ And Celeborn said: ‘I do not know, Eldest.’ But Galadriel said: ‘Not in Middle-earth, nor until the lands that lie under the wave are lifted up again. Then in the willow-meads of Tasarinan we may meet in the Spring. Farewell!’
This seems to suggest that unless something else causes it, the Ents do not die.
Note on immortality
Throughout the main body of the answer I've been careful not to blindly use the term immortality. In Tolkien's Legendarium, immortal beings are tied to the Earth and live and die by it. Even if an Elf's body is destroyed, they return to the Halls of Mandos and can be "reclothed" to return to Arda physically. This is however not known about Ents. After Beechbone's death in the Battle of Isengard it isn't suggested that his fëar will simply be transferred to another tree. He is presumed to die. In that case their fëar is not tied to Middle-earth, and like Men passes on at death. So while they may be incredibly long lived they may not be tied to Arda in the way Elves are. None of this, however, is certain. It may be that as an Ent's fëar can move between Olvar (trees/flora), although Treebeard's suggestion that the Ent's are dying out would go against that.
Based on all the evidence (what little we have) I think it is very unlikely that the Ents are immortal in the same way as Elves. While they only died from physical harm done to them and some spirits can be trapped by their body becoming tree-like. If an Ent is to die, their spirit would not be repurposed and continue to live on but would return somewhere either to Aman or out into the Void.