I know that some solvents will remove tape. I believe that there are things that work at taking off sticky labels from picture frames: I'm not sure if it's vinegar, or alcohol or Windex. You want to make sure it doesn't damage the finish on the board. I have found, however, that sometimes you can use a stronger tape to remove weaker tape. Duct tape might work: it sure is sticky. I also use Scotch brand (clear) mailing tape. I love this stuff. It really keeps things securely closed. (I even use it to close cereal bags, as long as I fold down a corner in order to have something to grab onto that will let me pull it off the next time.) Try using a strong tape. Press it onto the old tape, even rubbing it on hard with your nail or finger. However, sometimes peeling things off SLOWLY will make for a better result.
You can use that double-sided tape again this year, but fold one end down on itself, just enough so you can grab it with your nails when you're ready to remove it. There is also a blue sticky subtance with a duck on it, and I think it's called DAP. It has to be the right brand. (Sometimes Walmart carries it, or you can find it on line.) It is very good at attaching papers to a white board and you can peel it off and use it again. Schools (in my experience) order the stuff that never works.
#10 mentions Goo Gone. Good idea. I think there may be something else called Goop Off. And you might try olive oil, too.