I was not able to find anything as to how long it lasts, and I think it might not go away at all. However, I did find another way to cure it. Hayfever Pills.
Gifts from random events, found in the ground, dropped by wild Pokémon - Serebii
As far as I know, traveling merchants also sell Hayfever pills. Hayfever Pills will cure Hayfever, and are a consumable item.