I am a fairly new player but in my gameplay i have encountered this problem several times. On completing a quest which requires you to collect an object. If your inventory is full you cannot view or collect the object concerned causing the quest to fail. On trying to complete that area is greyed out resulting in failure. I have tried abandoning and requiring the quest ensuring inventory space without success, if there is an easy answer please enlighten me.
That has always been the case, ever since the launch of WoW. If you are lucky, the item will be mailed to you in the event that your inventory is full. But in the end, monitoring inventory space is on the player, not Blizzard.
In other words, this is not a bug, and fully intended.