Theories with Fatal Flaws
One of the strongest constraints on Bombadil's nature comes from Glorfindel's words at the Council of Elrond. Objecting to Tom as a guardian of the Ring, Glorfindel says,
"...soon or late the Lord of the Rings would learn of its hiding place and would bend all his power toward it. Could that power be defied by Bombadil alone? I think not. I think that in the end, if all else is conquered, Bombadil will fall, Last as he was First; and then Night will come."
Unless Glorfindel is flat out wrong, this makes it clear that Bombadil is weaker than Sauron in a direct conflict of "power," whatever that term means (it might well include Sauron's full military strength). Galdor concurs, saying that
"Power to defy our Enemy is not in him, unless such power is in the earth itself. And yet we see that Sauron can torture and destroy the very hills."
Galdor admits to knowing "little of Iarwain save the name", so it must not take great learning to make general statements about his power and perhaps its source. Significantly, nobody at the Council objects to either of these statements. It seems likely that whatever limited knowledge Galdor based his comments on was common among the wise and learned Elves at the Council. More specifically, Elrond is clearly reasonably familiar with Bombadil and Gandalf seems to know quite a bit about his nature and abilities, but neither of them object.
Although it is exceedingly unlikely that so many knowledgeable individuals at the Council were mistaken, some have suggested that they intentionally concealed the truth. This cannot be disproved, but it would be very different from the treatment of other secrets at the Council. For example, when Gloin asks about the Three Rings, Elrond states that "of them it is not permitted to speak", but nevertheless violates that prohibition (to a small degree). He does not feign ignorance or even simply remain silent: this Council does not seem to have been a place for partial truths.
Given this, it seems very reasonable to accept the Council's statements. This choice is corroborated in Letter #144, where Tolkien says that "Ultimately only the victory of the West will allow Bombadil to continue, or even to survive. Nothing would be left for him in the world of Sauron." While it does not speak directly of a conflict between the two, this quote makes it clear that Bombadil would be in some sense "killed" if Sauron was victorious, just as Glorfindel said.
The first and clearest casualty of this conclusion is the theory that Bombadil is Iluvatar. Tolkien states this directly Letters: in Letter #181 he says that "There is no 'embodiment' of the Creator anywhere in this story or mythology", and in Letter #211 he states that "The One does not physically inhabit any part of Ea." As if that weren't enough, Tolkien describes Tom's interests and motives in many places, but the motives of the all-powerful Creator probably could not even be expressed in words. All this makes the Eru theory probably the least likely explanation for Bombadil that still arises regularly.
The constraint from the Council is much stronger than this. The "Valaquenta" says that eight of the Valar "were of chief power and reverence":
"the Aratar, the High Ones of Arda: Manwe and Varda, Ulmo, Yavanna and Aule, Mandos, Nienna, and Orome. Though Manwe is their King and holds their allegiance under Eru, in majesty they are peers, surpassing beyond compare all others, whether of the Valar and the Maiar, or of any other order that Iluvatar has sent into Ea."
This includes all of the Valar whom I have seen suggested as identities for Bombadil except for Tulkas. I cannot believe that any of these great powers would be so weak (in any sense) that "the power to defy our Enemy is not in him". The gulf in power and majesty between the Aratar and even the greatest of the Maiar is "beyond compare": I do not believe that even Sauron and all his armies could "kill" Manwe or Aule, for example. (Although Tolkien wrote that "Sauron was 'greater', effectively, in the Second Age than Morgoth at the end of the First", that was because Morgoth alone among the Valar had expended his native power to gain control over the physical world. No "unfallen" Vala would do such a thing.) On this basis, I do not believe that Bombadil could be any individual on this list.
The theory that Bombadil is Tulkas escapes this argument, but it suffers from a fatal personality mismatch. According to the "Valaquenta", Tulkas "delights in wrestling and contests of strength" and "his weapons are his hands". Bombadil's only weapons appear to be his songs; he shows no interest in physical prowess. Also, Bombadil fights "evil" only when he walks into it or is called, precisely the opposite of Tulkas who came to Arda specifically to battle Melkor. While it is conceivable that his personality could have reversed in these ways, there is no reason to think that it did.
There is other evidence that Bombadil is not one of the Valar. In "In the House of Tom Bombadil", Tom says of himself, "He was here before the Kings and the graves and the Barrow-wights. When the Elves passed westward, Tom was here already, before the seas were bent." Bombadil says that he "was here already", not merely that he "had been here": this implies a long term presence in the area; in fact, this passage gives the impression that he never left. However, we read in The Silmarillion that after the destruction of the Lamps, "the Valar came seldom over the mountains to Middle-earth". (The only Valar said to spend much time there were Yavanna and Orome, and apart from Orome's time with the Quendi before the Great Journey these seem to have been brief visits rather than extended habitation.) While falling short of a complete proof on its own, this strongly implies that Bombadil is not one of the Valar.
Another argument is that the Valar were the governors of Arda and deeply concerned with the fate of the Children of Iluvatar (only during the Akallabeth are we told that "the Valar laid down their government of Arda"). However, when asked if Bombadil could guard the Ring, Gandalf says that, "He might do so, if all the free folk of the world begged him, but he would not understand the need." Even in the unlikely event that a Vala abandoned his responsibilities and chose to ignore all of the evil (and good) in the world, it could never be said that he would not understand the struggle.
While I believe that the arguments above are more than sufficient, I should also point out a basic difficulty with the most common Bombadil-as-Vala identification, the claim that Bombadil and Goldberry are Aule and Yavanna. Later, I discuss the strong connection between Goldberry and water, particularly the river Withywindle. This would be very surprising for Yavanna, whose cares far more for plants and animals than for rains and streams. Goldberry does not show any particular interest in living things (except water lilies). For his part, Bombadil shows almost no interest in craft or building and a great deal of interest in living things, entirely contrary to Aule's character. If Bombadil had to be one of the Valar, Aule might be the best fit, but he does not, and the fit simply is not that good. [Does this argument still seem insufficient, maybe based on other things you've read? Here's a full essay on why Bombadil can't be Aule.]
Finally, we can safely conclude that Bombadil did not belong to any of the "major races": Elves, Dwarves, or humans (including hobbits). The fact that the Council of Elrond even considered whether Bombadil could hold off Sauron's full power seems proof that he was not some mere Elf or human, however strong. Finrod, a Noldo of Valinor, held out against Sauron briefly when he was captured with Beren, but even he eventually succumbed. And then, Sauron was seeking information, not trying to slay his captives outright (which proved much easier). Bombadil himself provides an even stronger argument when he tells the hobbits that "When the Elves passed westward, Tom was here already": he was there before the Elves arrived, and they were awake before all other sentient races in Middle-earth. (It goes without saying that Bombadil is not an Ent, Troll, or other such race: in addition to the failings above, they were clearly non-human.)