For sure, some of the poems merely set the mood, while other convey background information, though I think you can get by without reading most of them. (This will however reduce your enjoyment of the book — they're there for a purpose after all.)
The Fellowship of the Ring, book one: hobbit and elvish lore
I don't think there's any knowledge to be gleaned from “The Road goes ever on and on”, for example. This is Bilbo's parting poem in 1.II, and there are two variations: 1.III, said by Frodo when he in turn leaves the Shire; 6.VI, said by Bilbo, just before returning to the Shire. This echoes a similar poem (“Roads go ever, ever on”) in The Hobbit (19 (last chapter), said by Bilbo as he returns to the Shire). But comparing the different versions reveals or at least confirm a subtext (I hadn't really thought about it until seeing the citations in the Wikipedia article).
At the other extreme, the rings rhyme is absolutely crucial to the plot (1.II: “Three Rings for the Elven-kings under the sky”, best known for the inscription on the One Ring: “One Ring to rule them all, One Ring to find them, One Ring to bring them all and in the darkness bind them”). But it's explained in the accompanying text anyway.
There are many other hobbit rhymes and songs that set the mood. I think they mainly show the hobbit's earthy, homely nature while revealing a streak of wistfulness for adventure. They are not specific to Middle-Earth — they may be sung by hobbits, but they could equally be human folklore.
- 1.III: “Upon the hearth the fire is red” (before meeting Gildor)
- 1.IV: “Ho! Ho! Ho! to the bottle I go”
- 1.V: “Sing hey! for the bath at close of day”
- 1.V: “Farewell we call to hearth and hall!”: Merry and Pippin's somewhat naive idea of adventuring
- 1.VI ”O! Wanderers in the shadowed land”
- 1.IX “There is an inn, a merry old inn” (sung at the Prancing Pony in Bree), best know for the cow jumping over the moon (at which point Frodo dons the ring, causing much ado, and goes to speak with Aragorn, then known as Strider)
- 2.III “When winter first begins to bite”
- 2.III “I sit beside the fire and think”
- 4.III “Grey as a mouse” (the Oliphaunt rhyme)
The Fellowship of the Ring, book two: more lore
Our first encounter with elves is with the song “Snow-white! Snow-white! O Lady clear!” (1.III). This song introduces the name of Elbereth (“Gilthoniel! O Elbereth!”) and the idea that elves long for the West, but most importantly it's the Elves' entrance song.
Tom Bombadil's songs “Hey dol! merry dol! ring a dong dillo!” (1.VI ff.) tells us who he is — a mysterious figure living with his wife close to nature and without a care in the world.
I don't think “Cold be hand and heart and bone” (1.VIII, the Barrow-wights' song) is telling us anything about the world of LOTR. This is just the dead frightening the living.
The rhyme in Gandalf's message to Frodo received in Bree (1.X, “All that is gold does not glitter”) is fairly obviously conveying a message to Frodo (“Not all those who wander are lost”, and so on, describing Aragorn), and it's hardly a stretch to realize there's information for the reader (“A light from the shadows shall spring; / Renewed shall be blade that was broken, / The crownless again shall be king”). Later, during the Council of Elrond (2.II), we hear Bilbo's version.
On the way between Bree and Rivendell (1.XI), Sam reveals some elven lore: “Gil-galad was an Elven-king”. The surrounding text tells more of Gil-galad's place in history than the song.
A little later, Aragorn tells the tale of Beren and Tinúviel (Lúthien) (“The leaves were long, the grass was green”). While this tale does not directly impact the main plot, it is important background for Aragorn and Arwen, as Beren and Lúthien are the point of reference for love between mortal humans and quasi-immortal elves.
The troll song (1.XII, “Troll sat alone on his seat of stone”) warns both the characters and the reader that trolls are tough, in addition to being nasty. Not that you wouldn't guess.
In Rivendell (2.I), Bilbo tells the tale of Eärendil (“Eärendil was a mariner”). There's some history in there, especially about the Silmarils, as well as a lot of nostalgia. I don't think any of it is directly relevant to LOTR though.
Then (2.II) Boromir tells of a message in verse that he heards in a dream: “Seek for the Sword that was broken”. This is obviously a prophesy from the context, and the characters explain it in detail.
In Moria, Gimli sings a tale of the time of Durin (2.IV, “The world was young, the mountains green”). There is some information about dwarven lore there, but not essential to the plot of LOTR. Gandalf explains the relevant points to the hobbits — mainly expounding on mithril.
In Lothlórien, Legolas tells the story of Nimrodel (2.VI, “An Elven-maid there was of old”). We learn in passing that she was supposed to board a west-bound “elven-ship in haven grey”. The tale is a pretext for Legolas to explain the hostility of elves towards dwarves.
Later, Frodo and Sam compose a song in memory of Gandalf's (apparent) passing in Moria (2.VII, “When evening in the Shire was grey”). This song is sadder and more profound than their earlier light-hearted folk songs, showing that they are already maturing from their travels — Frodo rather more so than Sam.
As the party leaves Lothlórien, Galadriel sings of her nostalgia for Ilmarin in the West and the Trees of old, and of her fears for Lóthlorien.
(2.VIII “I sang of leaves, of leaves of gold, and leaves of gold there grew”) I don't think the allusions there are built upon elsewhere in LOTR. She also gives a hint of Frodo's eventual journey West in her parting speech “Ai! laurië lantar lassi súrinen” (“Maybe thou shalt find Valimar.”).
The Two Towers, book three: human lands and Ents
Boromir's eulogy (3.I, “Through Rohan over fen and field where the long grass grows”) gives a little insight on geography as perceived by Aragorn, Legolas and Gimli: to the North are “the empty lands where no men are” the Sea to the South, towards which the Anduin flows; mountains to the West, and of the East one does not speak.
As Aragorn nears Gondor, he briefly sings his longing for it (3.II, “Gondor! Gondor, between the Mountains and the Sea!”). We learn of the greatness of kings past, and Aragorn expresses his yearning for the light of the Silver Tree.
Treebeard's rhyme (3.IV, “Learn now the lore of Living Creatures!”) tells us a little about how ents perceive the world: the four peoples are the elves, the dwarves, the ents and the men (in that order). Merry and Pippin suggest to add a line for hobbits, which he will do (3.X). Later, Treebeard sings of his travels (3.IV, “In the willow-meads of Tasarinan I walked in the Spring”) — to Tasarinan, Ossiriand, Neldoreth, Dorthonion, which would reveal something about his age (many of these are First Age places) and the extent of his travels, if the reader was able to place the names (Tasarinan is described as one of the “lands that lie under the wave” (6.VI, a); that's not a reference you can understand from LOTR alone). Treebeard also relates an elvish duet about the Entwives (3.IV, “When Spring unfolds the beechen leaf, and sap is in the bough”). That song prophesies that the Entwives may come back “when Winter comes, and singing ends”, and that the Ents and Entwives “together […] will take the road that leads into the West”: so if there are no Ents in forests near you, maybe they did find each other and walk away. Bregalad also sings a few lines (“O Orofarnë, Lassemista, Carnimírië!”), of no plot significance, like the Ents' marching song (“To Isengard! Though Isengard be ringed and barred with doors of stone”).
Galadriel's messages to the company (3.V, “Where now are the Dúnedain, Elessar, Elessar?”, “Legolas Greenleaf long under tree”) are prophetic as one might expect. A keen re-reader will understand that she is telling Aragorn that he must raise an army from the Paths of the Dead and Legolas that he will soon sail West (the recipients will take a while to cotton on).
The Rohirrim's rhyme (3.VI, “Where now the horse and the rider? Where is the horn that was blowing?”) tells us that even this seemingly simpler people (the free-ranging horse-riders) know nostalgia. When Gandalf stands before Théoden, he praises Lothlórien (3.VI, “In Dwimordene, in Lórien”). The reader won't learn anything new. Neither will he from the Rohirrim's call to arms (3.VI, “Arise now, arise, Riders of Théoden!”; a variant appears in 5.V). When the Rohirrim are puzzled by the sudden apparition of a forest (3.VII), Gandalf cryptically alludes to Ents (3.VIII, “Ere iron was found or tree was hewn”), but the reader knows about them already. There is one more marching song in this episode of which but two lines are cited. (3.IX, “Though Isengard be strong and hard, as cold as stone and bare as bone”).
The Two Towers, book four: a grim journey
Frodo's journey towards Mount Doom is a somber tale, little interrupted by poetry.
Gollum's rhymes (4.II, “The cold hard lands”, “Alive without breath”) are character characterization rather than people characterization. His harsh, barbaric nature contrasts with that of the easygoing hobbits (4.III, Oliphaunt rhyme, see above).
In the dark, when Frodo is incapacitated by Shelob, Sam is briefly possessed to cry “Gilthoniel A Elbereth!” (4.X), In Cirith Ungol, Sam sings a hobbit tune once more, but with new words, more elvish that a hobbit's, that show how his adventure has transformed him (6.I, “In western lands beneath the Sun”).
The Return of the Kings, book one: Rohirrim lore in the making
We return to the Paths of the Dead, and Aragorn relates the words of Malbeth the Seer (5.II, “Over the land there lies a long shadow”). Well before Galadriel, he prophesied Aragorn's summoning of the Dead.
Unusually, when the Rohirrim ride East to fight Mordor, Tolkien relates a song that belongs to the future — a song that will later be composed about that war (5.III, “From dark Dunharrow in the dim morning”). He does this again about the battle of Minas Tirith (known as the Mounds of Mundburg in Rohan) (5.VI, “We heard of the horns in the hills ringing”).
I believe this song includes a few details about who took part and how the journey and battle proceeded that are not given out in prose. Nothing major, however.
Théoden's eulogy on the spur of the moment is brief (5.VI, “Mourn not overmuch! Mighty was the fallen”). There's still a battle to be fought at that point. For completeness, I'll mention his horse's (5.VI, “Faithful servant yet master's bane”). Similarly brief is Éomer's fighting couplet (5.VI, “Out of doubt, out of dark to the day's rising”). We'll hear a few more lines of eulogy for Théoden later (6.VI, “Out of doubt, out of dark, to the day's rising”).
The old wives' rhyme about kingsfoil (5.VIII, “When the black breath blows”) illustrates the decadence of Gondor — a healing herb reduced to a rhyme, though this is understandable as the herb is only potent in the hands of a king, which Gondor has not had for a long time.
Legolas sings of Lebennin past (5.IX, “Silver flow the streams from Celos to Erui”). It is hardly surprising that this region of Gondor is not as green as it used to be. More important to the plot is that Legolas heard the calls of the gulls and that, as predicted by Galadriel, they arose in him a yearning for the Sea.
The Return of the Kings, book two: parting words
As Frodo and Sam return from Mordor, they are praised in many languages (6.IV, “Long live the Halflings! Praise them with great praise!”). This doesn't tell us much, other than that there seem to be two different foreign languages (I haven't verified this).
After the battle is won, Legolas sings of his yearning for the Sea again (6.IV, “To the Sea, to the Sea! The white gulls are crying”). We do learn a bit of background about “the Lost Isle calling, / In Eressëa, in Elvenhome that no man can discover”. An Eagle comes and joins the singing (6.IV, “Sing now, ye people of the Tower of Anor”): Sauron is defeated (we've just read about that); and the King is coming and the Tree will be replanted (these two events will happen before the end of the chapter).
As Frodo is leaving for the Grey Havens, just before the hobbits meet Gildor again (see above), Sam sings one last time, but not a traditional hobbit song this time: “Still round the corner there may wait”, a short rhyme of adventuring (6.IX). Sam may be staying at home in the Shire, but he's known adventure. The elves respond with a final rendering of “A! Elbereth Gilthoniel!”.
Very little of the poetry matters to the plot: the One Ring rhyme and Gandalf's message are the only two cases that contain important information, and the both are explained in the surrounding dialogue. A few prophesies come true, and the keen reader will get a glimpse of the tales told in The Silmarillion. But mostly the poetry is there for characterization:
- Much of it illustrates the various peoples' cultures. The poetry is noticeably more abundant near the beginning, and when we meet new people. It's also somewhat telling we don't get Orc songs.
- Some of the songs illustrate the adventuring hobbits' mental journey.
So if you skip all the poetry, you won't miss any important background information. But if you do that, you might as well read the Cliff Notes and skip the book altogether.
Now, why don't people read the Appendices? (Ok, in part because so many publishers omit them, but why?)