When writing, how should I describe a member of a humanoid species, like an elf, who looks Asian? I can't describe them as Asian because there's no Asia or even Earth where they're from but I can't describe them as yellow because that's rude.

It's not rude to describe an elf as yellow. Elves can be technicolour if you like -- they're your creation. What about some shade of golden skin texture?

Since, in your story, you don't have an Asia to compare them to you'll just have to choose Asian characteristics to give them and describe those in turn.