Elves, fae, and dragons don't all exist in the same lore, so the only war that they would have would be UPG or simply made up.
Elves and dragons exist in Norse lore as Alfar and (one instance of an actual dragon) as a Dvergr shapeshifting into a dragon out of greed over a cursed ring (story sounds familiar, right? Tolkien was a Germanic scholar). Despite what some draconic paths/SPGs state, Jormungandr is a serpent and not a dragon.
The word Fairy doesn't apply to one specific sect of spirits as it is applied to a group of multiple types of spirits. However, the term really is only applicable to, as I understand it, Celtic spirits. Even then, the traditional lore doesn't calle them Fae (and there is one mention of elves in Celtic lore, but I've not studied that venue in-depth).