According to Good Housekeeping, the best way to remove sticky residue from clothing is to use an acetone-based cleaner. Place the clothing sticky side down on a towel and pour some acetone or nail polish remover onto the clothing. Place another towel on top of the clothing and rub gently for a few minutes to loosen the residue. After this, it should be safe to machine wash the clothing normally.
How to Clean Stuff explalins that a number of other cleaning products work in lieu of acetone, including dishwashing soap, peanut butter, WD-40, vegetable oil, Avon Skin-So-Soft lotion, Goo-Gone, and rubbing alcohol. Be careful when choosing an alternative cleaning product, as many of these substances have the potential to stain while removing the adhesive residue.
How to Clean Stuff suggests using a toothbrush to clean the sticky residue away if rubbing between two towels does not work. Additionally, make sure that the residue stain and any stain left from the cleaning product is gone prior to placing in the dryer. Heat sets the stain and makes it more difficult to remove. For fabric that is dry clean only, use a professional dry cleaner instead of trying to remove the stain at home.