My daughter is in college. Like most college students, she and her suitemates like to hang things from the ceiling. They can't use nails or tacks on their white walls and ceilings because they leave marks and holes.
I think it's 3M (red and yellow packaging) that makes removable hooks and stickers. These are fantastic! They have these small clear hooks that would hold the fishing line perfectly, and they remove cleanly if you decide to move them later.
They worked great in the dorms last year. You can even hang the larger hooks on the wall for light jackets and things. I've seen them at Target and Michael's, usually on an endcap or near tapes and hooks.
I think this is what you are looking for. Good luck!