or does it permanently disappear?

thats a good question

you have to do it.

there are some where you can trash and not have to ever do like the crafting skills..

if you choose to do all of them, but then later find out you dont want to collect all the mats,

just click trash. and they will later appear at the elder

It doesn't permanently dissappear, unless of course it was a quest you can only do at a certain level.
But most, you can just go back to the person who gave it (which i do a lot because i hate a cluttered quest box)

this. lol i am so awkward when i type something... xD

blackice put it brilliantly xD