I know the drop box for the illum gree event stays after the event however the BBA and recently the Rakghoul event drop boxes disappear. If there are other locations to hand in those quests that i don't know about feel free to let me know. Some people might say that we should hand in our quests before the event ends as the end date is posted however with the rakghoul event a message appeared saying we would be shuttled to fleet in 10 sec before reset time for dailies which is when most people expected it to end. I believe that it would be a good idea to put a general event drop box somewhere on fleet to enable people to hand in quests after events have finished or even during, its annoying to travel to the fleet and then remember you forgot to hand in the quests.