Work a plastic knife or paint scraper underneath the tape and residue to carefully peel the tape away from the carpet.
Douse a sponge or clean cloth with dry cleaning solvent or rubbing alcohol. Blot the affected area until the solvent completely transfers to the carpet. Solvent and alcohol dry quickly, so apply generously.
Pour 2 cups of warm water into a non-metal container. Add 1 tbsp. of white vinegar and 1 tbsp. of dish soap. Apply the solution to the stain with a clean sponge until the liquid absorbs into the stain.
Rinse by sponging the area with cold water. Dry with an absorbent towel.
Test carpet colorfastness. Snip a single fiber from the carpet, or test inside a closet or similar hidden area. Spray the sample area with WD-40, as recommended by Duck brand products, or another petroleum-based lubricant.
Return 15 minutes to half an hour later. Examine carpet for fading.
Spray the stain with WD-40 if the carpet passes the color test.
Scrape duct tape residue as it lifts and dissolves with a plastic knife or paper towel.