I am not sure if this is true, but I heard there were short poems in the beginning of each chapter in Lord of the Rings. Although, this could be done fairly easily, I am wondering if there are any other way to incorporate poems in a novel. I am thinking there are many instances of it in the rich history of literature, but because I haven't read a lot of books I am curious to know if there are ways of enriching a novel with poems that I am not aware of.
You have been misinformed: The Lord of the Rings doesn't have short poems at the start of each chapter. The Lord of the Rings has poems of various length (up to several pages long), when characters sing, recite poems, or find them written somewhere.
Characters may sing on varied occasions: there are walking songs and bathing songs, there are elegies for the dead and lays sung to tell a tale. Verse may be recited, etc. Whenever the narrator says a character sings, or recites poetry, or encounters verse in some other form, the verse is right there. It's as simple as that - the verse is as much a part of the narrative as any bit of dialogue.
Such use of poetry is quite common in literature, if the author can write poetry.
@Rasdashan mentions Dr. Zhivago in a comment. Boris Pasternak uses a different approach: while the main character is a poet, only snippets of verse appear within the narration. Instead, a collection of verse "written by the character" is appended to the book. That approach is more meta, keeping up the pretence that the character is a real person.
And of course there is the separate form of writing a novel entirely in verse. Such were the ancient sagas, such as Beowulf and the Iliad, medieval works such as El Cantar de Mio Cid and The Song of Roland, and more modern novels in verse, such as Alexander Pushkin's Yevgeniy Onegin.
There are a number of short to moderate verses in The Lord of the Rings They do not serve as chapter headings, nor do the appear regularly in each chapter. They are generally recited or sung by one or another character, although in at least one case the narrator tells us that a verse was made years later to describe and commemorate an event, and gives us the verse. In another case the narrator describes an epitaph that was later added to a gravesite, after the scene where the death occurs (Faithful servant / yet master's bane / Lightfoot's foal / swift Snowmane for Theoden's horse). Verses are recited or sung by Aragorn, Gimli, Legolas, Frodo, Bilbo, and Sam, among others. That is one way to do it.
I have seen works that use verse as chapter epigraphs, and works where "ancient" verse is quoted by the narrator. It all depends on the narrative situation, and the effect desired.
Any use of verse takes some care. If the tone and style do not fit, or the composition seems clunky, it will detract rather than add.
I recall that in Delany's Babel-17 the main character is a poet, and a number of her verses are quoted. But then Delaney was at that time married to the noted formalist poet Marilyn Hacker, and i rather suspect that she had a hand in those verses.
There are answers before mine; so I will only add that one other use of a poem or rhyming verse (which I did not see in those answers) is as a clue to some mystery. This can represent a riddle directly, or it is a riddle disguised as poem, like a love poem or an elegy on a gravestone.
I cannot recall exactly, but I believe Dan Brown has used this device in his religious mysteries.
Two novels that do this very well are The Translator, by John Crowley, and Possession, by A.S. Byatt.
The Translator is about a young American poet who translates the work of a great Russian poet who is a visiting professor at the college she is attending. I believe I read somewhere that John Crowley wanted to write a novel about a great poet, but didn't think he could write great poetry himself. So the great poetry is in Russian, but all we see is a fairly literal free-verse translation into English. The young American poet becomes a celebrated poet herself after graduating from college, but mostly we see her earlier works.
In Possession, Byatt has a present-day plot and a Victorian plot. The Victorian plot is about two poets, so she had to write poetry for it—and to it had to be fairly good poetry to maintain the reader's suspension of disbelief. Luckily, she was a good enough poet to pull it off.