Depending on the surface, the most thorough way of removing double-sided tape residue may be to scrape it off. Use a clean razor blade, and carefully wedge the blade under the residue, beneath the foam portion of the double-sided tape or its residue. Be careful not to scratch or gouge the surface. Gently scrape forward, away from yourself, to remove the residue. Repeat the process until the residue is completely gone. If you are working on a non-porous surface, like glass or ceramic, use window cleaner and a rag to remove any marks left behind. Because it's sticky on both sides, double-sided tape can quickly gum up your razor blade. If this happens, use window cleaner or a citrus-based cleaner to remove the foam pieces or sticky residue from the razor blade as you work.
Chemical cleaners like Goo Gone, Goof Off or other adhesive removers are excellent for removing double-sided tape residue. Be careful, however, that the cleaner does not damage the surface. If you are working with wood or other porous surfaces, test an inconspicuous area before beginning. Simply spray the cleaner on the residue, let it sit for a moment, then rub it away with a clean rag. You may need to use a fingernail or razor blade to scrape any stubborn remaining spots, or try a repeat application of the cleaner. If pieces of double-sided tape remain intact, you may need to give the cleaner a way to penetrate through the outward facing sticky side. You can either carefully use a razor blade to scrape away the exterior side, or use the blade to score the tape so the cleaner can soak through.
Steam will often lift double-sided tape residue right off the surface without any elbow grease required. Be sure that the surface you are cleaning will not be damaged by water or the heat from the steam. If pieces of tape remain, remove the outer facing sticky portion by gently scraping it off. You can also use a razor blade to either score the tape, or a wallpaper scoring tool to cut through the tape and allow the steam to get through to the other side. Using a steam cleaner, steam wrinkle remover or steam wallpaper remover and direct the steam on to the residue for several seconds. Let the residue cool slightly, or wear gloves, and then pull it away from the surface. If you don't have a steam machine, and the surface you are cleaning isn't too big, you can boil a pot of water and then set the item on a baking rack over the pot of water to let the steam do its work. You may need to gradually scrape off the exterior sticky residue of the double-sided tape layer by layer as the steam melts and loosens it.