[mhagain]

Treebeard is one of the "Shepherds of the Trees" and they can be definitively dated to the time the Children of Ilúvatar awoke:

Behold! When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared.

Roundabout now it's necessary to mention Gandalf's words about Treebeard:

...he is the oldest of the Ents, the oldest living thing that still walks beneath the Sun upon this Middle-earth.

The first observation here is that "oldest of the Ents" need not necessarily mean one of the firstborn Ents, who could have all died out (or gone tree-ish) by the end of the Third Age.

The second observation is that Gandalf is claiming Treebeard to be older than Círdan, which is quite important, as Círdan is almost certainly the oldest Elf left in Middle-earth, was on the Great Journey, and may even remember Cuiviénen. This therefore dates Treebeard's age to at the time the Elves awoke, or very shortly after it.

The third observation is that Gandalf is drawing a distinction between "living things" and spirits; Gandalf as we know is one of the Ainur, he existed before the world was created, and is therefore older than Treebeard by default, so when Gandalf says "oldest living thing" he is not including the Valar, the Maiar and other spirits.

Bombadil seems to claim to remember a time when most of Middle-earth still lay under the Sleep of Yavanna:

Mark my words, my friends: Tom was here before the river and the trees; Tom remembers the first raindrop and the first acorn.

However that's a fairly wide spread of time and could date him to any time from the fall of the Lamps to the first rising of the Sun. Depending on how you interpret his words, he could even be referring to the First Spring of Arda, i.e the time of the Lamps:

Then the seeds that Yavanna had sown began swiftly to sprout and to burgeon, and there arose a multitude of growing things great and small, mosses and grasses and great ferns, and trees whose tops were crowned with cloud as they were living mountains, but whose feet were wrapped in a green twilight.

A lot of this hinges on the related question of who Bombadil is.

If Bombadil is an older class of spirit, or even one from before the world was created, then his claim to remember the first acorn puts him at the earliest end of this possible timescale, which makes him older than Treebeard. However, we don't know who his is so he could just as easily be at the later end of it and be referring to the Second Spring of Arda, i.e after the Sun rose, and he's therefore possibly younger than Treebeard.

However again, noting that Gandalf exempted spirits from the class of "living things" above, we have to wonder if the question of which was oldest even makes any sense. In the end we just don't have enough information to give a definitive answer.