Do the Fey have an enemy race?

Question

Do Fey have a particular race or ideal they dislike?

Answer 1

Traditionally, that would simply be other fey. Fey are often divided into two or more “courts,” usually “seelie” or “summer” vs. “unseelie” or “winter,” though going with a court for each of the four seasons is not uncommon either. The Courts Seelie and Unseelie were used in Dungeons & Dragons, including the 3.5e that Pathfinder is based on. Seelie is “good” and unseelie is “evil,” but these terms mean different things to fey than they do for celestials and fiends. They are much more about the natural forces of growth and decay than they are about morality, and in any event their fey minds are near-completely alien to mortals’. Fey Courts are often highly political, as the name might suggest, and in D&D the courts have a strong presence in interplanar politics.

However, Golarion specifically eschews the fey courts, instead having the First World of the fey “ruled,” as much as such a word applies to the fey, by the “Eldest.” The Eldest are like demigods, and have a lot of ability to shape the First World so their realms are similar to the divine realms of the gods. But most of the First World isn’t really under the sway of any Eldest in particular. The First World doesn’t engage with planar politics, and so the concept of an “enemy race” does not really apply to them.

Answer 2

This DM runs the PathFinder ruleset in the old AD&D2E Birthright setting. You did ask about other settings...

In that setting the Ethereal Plane is a blend of two planes further influenced by a 'dead' god.
The traditional Ethereal (Border and Deep), and an less pronounced equivalent of "The First World" are there but they are being corrupted by the influence of the Cold Rider (an aspect of the dead god Azrai) wherein the plane, especially the Border Ethereal, is becoming like the near Umbra of the White Wolf setting. In the "Shadow World" (Border Ethereal) of Birthright, pockets of stability form around areas that have intense emotional history in the real world and these stable areas tend to be static enough that the undead find and inhabit them. When the veil between the sunlit world (Aebrynis) and the Shadow World thins (usually Aebrynis' winter) thin patches open up in the Evanescence (veil) and the undead push through. Undead are the Cold Riders' troops of choice.

In the ensuing conflict between what passes for a First World Fey in the Birthright setting (they have a different history entwined with the Sidhe of Birthright) the Fey's enemy is "The Shadow" (Cold Rider aka Azrai) and all his forces.

This plays off the opposing natures of the traditional Ethereal/First World which is mutable/changeable and dream-like, and the dark, banal, static, corruption of the Cold Rider (haunted near Umbra).

So for me it is Fey vs Undead.
A conflict made difficult for the Fey because much of their enchanting/beguiling nature, and illusion trickery, is much less effective on undead.