Are you required to complete each day in your task list?

When you check off items in your task list, you will need a star to complete it. These stars are obtained whenever you complete a puzzle level.

However, if you do not want to complete a day yet, it's okay to do so. You can stock up on the stars as long as you want and just play the puzzles, you can complete the task list later on if you have enough stars that you are content with.