The method used by the followers of Sauron to "activate" Mt. Doom is based in real science. "Phreatomagmatic eruptions" occur when water is introduced to hot magma. The rapidly changing pressures and temperatures of the water and magma eventually lead to a volcanic eruption. That said, the eruption in the show occurs much faster than any natural process would probably take.
The command Galadriel gives to her mount while pursuing Adar on horseback, "Noro lim", is the same command Glorfindel and Arwen give to their horses in the book and film versions, respectively, of The Lord of the Rings: The Fellowship of the Ring (2001).
When the Orc Bazur orders the other Orcs in Black Speech to search Ostirith, the word he uses for "Find them" is "gimbatul", as in "ash nazg gimbatul" = "One Ring to find them".
The words Bronwyn spoke to Theo when he had bad dreams as a boy are mostly an extremely close paraphrase of a passage of Sam Gamgee's interior monologue in J.R.R. Tolkien's "The Return of the King": "...in the end the Shadow was only a small and passing thing: there was light and high beauty for ever beyond its reach."
On the ship looking for the first sight of land, Isildur says to Galadriel "Keen are the eyes of the elves" after she tells him it has been visible to her for nearly an hour. In the book by JRR Tolkien, Lord of the Rings: The Two Towers chapter two, Aragorn, Isildur's descendant, says the same thing when Legolas is able to see details of the Riders of Rohan from five leagues away.