But we should probably begin with Finnish. Tolkien encountered it early in his career and retained a great fondness for it, drawing from its sound and structure as he shaped his elven speech. If you compare the sound only -- listening, say, to Tolkien's recitation of Namarië and a reading from the Kalevala (here's one -- go to 3:25 to hear the actual reading). This is the Finnish national epic, first published in book form in 1835, centers on the story of Väinämöinen, a sage who sings things into being (shades of Tolkien's Ainulindalë!). It was collected from oral traditions, and a greatly expanded version was preserved in the 1840's -- just in time, as the last who could recite its verses were just then passing from the world.
Of course, others of the languages Tolkien drew from also had their traditional epics -- for the Irish, it was the Táin Bó Cúailnge (the cattle-raid of Cooley), which you can hear passages from here. The Welsh have their Mabinogion, which includes some of the earliest version of the Authrian legends -- but, shame on them, there's not a single online recording of it in Welsh online (though you can hear a contemporary Welsh poet here). Compare either of these to Tolkien's A Elbereth Gilthoniel -- you can hear a recording I made of it in Sindarin here.
Many have speculated as to exactly how these existing languages influenced Tolkien's created ones; have a look at Alok Jha in The Guardian, Stan in the Sentence First blog, or Greg Harvey's piece at the For Dummies website for starters.
In the end, whatever his interest an any particular language, Tolkien remained a philologist at heart -- a lover of words, and tracer of all languages and their histories. He knew how many different kinds of events -- a split in population, migration, or emigration -- could affect the ways in which language changed. And so, to explain the changes in his own invented languages, he needed people and histories -- and this, in many of his later statements was the explanation he gave for why he created Middle-Earth.