Deleting Characters

Quest items don't disappear if they are stored in another class or bank and you delete the class they belonged to, so you would have to forcefully remove them from your inventory yes. But before doing that make sure they belong to your deleted class and not another class ;)

 

There is the old "fill your inventory with emeralds" method, in which you place your quest item in your inventory and then fill your inventory with emeralds or other stackable items, next you replace the quest item with an emerald, while still not leaving any empty inventory spaces, it will remove the quest item permanently.

For others reading I wouldn't recommend using these methods to remove quest items you need just for extra bank space, they will be necessary in the future, and while /fixquests works to get them back, it's better to be safe and keep them.