The only way to remove a quest from your quest journal is to complete it.
When you're given an assignment in school, if you don't do it, you fail. When you're given an assignment in Wizard101, you have the option to complete the assignment at any time in the future, but you must still complete it. Just complete the assignments you are given, it is the only way you will be able to progress through the story of the Spiral.
If you want the prize, you have to build the ladder rung by rung. There are no shortcuts.
Optional quests don't disappear, either. I've had a Crafting quest for several months. I don't intend to finish it, and it won't go away unless I do.
Note that you're not limited to four quests, as might be guessed by viewing the screen. Each page contains four quests, and the arrow keys will scroll to other pages.
