Removing any kind of adhesive is a challenge. Double-sided tape can be particularly difficult, since the stickiness on the "outside" makes it harder to work with.
Apply an adhesive-removing solution
Choose between rubbing alcohol, Avon's Skin-So-Soft, WD-40 or the commercial adhesive-removing solution Goo Gone. Apply the solution to the double-side tape. Let the solution sit for several minutes.
Peel back the tape
Start to peel the double-sided tape away from the surface to which it is affixed. As you peel, apply more of your chosen adhesive-removing solution to the underside of the tape, pushing it back slowly until it is completely removed.
Wash away the tape residue
Use a moistened cloth to rub and scrape away any remaining residue left by the double-sided tape. Clean the surface completely, if needed, and allow it to dry.