The concept of a spell being stored on a scroll, which can be used once and then vanishes, shows up in numerous tabletop and video game RPGs. From a game design perspective, the idea of a one-use spell is sensible. But considering that the whole idea of writing is to record information permanently, the idea that a single-use spell should take the form of a scroll of all things is kind of unintuitive. What's the history behind this idea? Is it based off of some old mythology or folklore, or was it a later invention?
Moreover, over the course of the novel, the King of Elfland uses all three master-runes to effect magical changes that encompass both Elfland and Erl. And once each rune is used, it is gone, not to be used again. After initially holding back, the King finally starts using the master-runes to try to prevent his daughter's departure from Elfland:
They rushed forward, he taking her hand; the Elf King lifted his beard, and just as he began to intone a rune that only once may be uttered, against which nothing from our fields can avail, they were through the frontier of twilight, and the rune shook and troubled those lands in which Lirazel walked no longer.
Dunsany was listed in Appendix N: Inspirational and Educational Reading, from the first edition AD&D Dungeon Masters Guide as one of the sources that contributed to Gygax's development of the game, so he was almost certainly familiar with The King of Elfland's Daughter.
Single-use scrolls were a feature of some of Fritz Leiber's Fafhrd and Grey Mouser stories (published from the 1940s into the 1980s). I don't have my books where I can reach them, so don't have publication dates for all of the stories, but one of them, "The Lords of Quarmall" (originally published in 1964, now found in the collection Swords Against Death), I recall being mentioned in a foreword or afterword as one of, if not the oldest of the stories, in order of creation, though not of publication -- and in it, Mouser had come into possession of a single-use. self-destroying scroll carrying a very powerful spell.
This story, at least, predates the inception of what came to be the D&D magic system.
Beyond this, the Vancian magic that's the core of the original system, in which a wizard completely forgets a spell once cast and has to re-memorize it after resting in order to cast it again, fits well with the idea that such power might be single-use in written, ready-to-use form, as well. If casting a spell erases it from the caster's mind, why wouldn't it also erase the written representation from the physical world?
In addition to what was already said, I would add some background that could possibly have influenced the use of spell scrolls as a source of one-use spells aside from modern literature.
Modern 'swords and magic' fantasy has many origins. A lot of it comes from novels (Tolkien's "the Lord of the Rings" and "the Hobbit" had huge impacts on the general picture people have of fantasy). Probably one of the sources that shapes our idea of fantasy the most, aside from literature, is the Celtic culture and theology that was prevalent in Europe before Christianity. The story of King Arthur is a great example, he is a Celtic-born prince, and the magic in the world is based largely (if not entirely) on Celtic mythology. In fact, most modern ideas of magic in western culture are largely rooted in Celtic mythology. Another great example is The Witcher series, which is deeply rooted in Celtic mythology.
That said, the Celtic view was that magic was redirecting or reshaping the energies that make up the world. Their practice of magic included materials that they believed had special properties, incantations and rituals, and yes, magical runes. They believed that certain patterns could invoke magic; consider The Witcher (mind that witchers themselves aren't a Celtic creation, even though the world is largely rooted in Celtic mythology). The witchers in this fantasy use runes to invoke 'quick' (though weaker in strength) magic.
The concept of runes could possibly be the origin of the 'spell scroll' in modern fantasy. For those unpracticed in magic in Celtic mythology, materials and chants won't be a tool on their belt, but runes created by someone who is, or the passive magic from charms might be. That said, if a portable rune exists that can evoke magic, and it isn't a permanent tool that keeps its magic, why not use paper? Mages would be much less likely to engrave a rune, that can only invoke magic once, into something like metal (or even leather for that matter) than inking it onto paper.