Peel away as much of the weatherstrip as possible.
Soak paper towels in white vinegar, and lay them over the adhesive. Let the towels sit in place for about 15 minutes.
Remove the paper towels, and scrub the adhesive with one of them.
Continue the process, letting the paper towels sit on the adhesive for additional 15-minute increments as needed until all the adhesive easily rubs away.