Did leaving out Tom Bombadil create any plot holes in The Fellowship of the Ring or any subsequent LOTR films?

I don't think that he leaves any holes in the plot, but taking him out does leave out a bit of lore I think. The scene where Tom Bombadil holds the one ring was important only to his background, not the quest of the fellowship. The fact that the ring had no power over him says quite a lot about how powerful he himself is. And yet, he spends his life in leisure. So the question is: why would a being that powerful spend his time singing to trees (not to mention banging a spirit)?

The answer, I think, is hinted at by Gandalf's actions after the ring is destroyed. Gandalf was one of the Maiar, lesser Ainur, the immortal beings created before time. Gandalf was believed to be the wisest of the Maiar, though he was called Olórin, or 'dreamer'. He was one of several Maiar who were sent to Middle Earth to contest Sauron. While in Middle Earth, they were known as the Istari, or Wizards. When the remainder of the fellowship is returning to the Shire, Gandalf leaves saying that he is going to go see Bombadil. And I think he says something about his job being done, though I don't have a copy of the book to check. My theory is that Tolkien was drawing a parallel between the two of them here. Tom Bombadil is a great spirit whose time is past, or possibly who never had a purpose. When one of the Hobbits, can't remember who, asks Goldberry who Bombadil is she replies "He is." There's a lot of debate about who Bombadil actually is, but my theory, based on this, is that he is the spirit of Ea, which is the name of all that exists. In Quenya, Ea means to be, so Ea refers to 'that which is', or 'the world that exists'. So no, nothing goes missing that pertains to the main narrative, but I think it leaves out a bit of the impressively extensive lore that Tolkien built around LOTR.

Leaving out Bombadil had several implications for the movie:

• Peter Jackson would have needed a new deus ex machina to rescue the Hobbits from the Barrow Wights. This was moot however because PJ simply removed the Barrow Downs--much to my dismay: I'd love to see what Jackson would have done with it. With his genius for evoking a location, the Barrow Downs would have been a thing to behold.
• The Hobbits needed another way of acquiring their ancient super-special magic daggers. Again this was waved away by having Aragorn simply give them these out of nowhere on Weathertop. The bit about the daggers being made to combat the wraiths (which allowed Merry to help kill the Witch King) is easy enough to ignore in a movie.
• Bombadil, and the proposal of giving the ring to him, was simply left out of the Council of Elrond. The deliberation of the Council were so ruthlessly chopped anyway that this hardly registers. Sheesh, they didn't even mention the dwarf rings, or Balin's expedition to Moria, where he had hoped to recover the last of the Seven--but, realistically, cuts had to be made for the movie.
• The scene with Old Man Willow had to be either cut or relocated. As it was, it was relocated to Fangorn where Treebeard rescues them from an anonymous huorn. Eh, too bad, but not much difference in my estimation.

Those are changes that were made necessary by removing Bombadil. I don't think it really left any plot holes, though; Bombadil himself was really nonessential to the story. As I mentioned, my greatest regret about it is that PJ didn't find some other way of working in the Barrow Downs. I would dearly love to see his vision of it.

The only reason I thought Bombadil needed to be included in the movie is because I thought he was the most significant being in Middle Earth. The Silmarillion hints that he was one of their "gods", or at least a minor deity. Tolkien liked his songs throughout the tale, and Bombadil seemed to be full of many. At least the scenes would have created a pleasant interlude, especially if the barrow wights scene had been kept also. At that point the hobbits, and audience, would have enjoyed a pleasant break from death and darkness.