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.