Harden stuck chewing gum with cold, then peel and lift the last residue out of the fabric.
Put the garment in a plastic bag (gum-side away from the fabric where possible) and place it in the freezer for an hour or two. For larger items, hold an ice cube in a bag against the gum until it hardens. Frozen gum turns brittle and loses its grip on the fibers.
While it is still cold and brittle, peel and scrape the gum off with a dull knife, spoon edge, or your fingernail. It should crack and flake away rather than smear. Work quickly before it warms and softens again.
Dab any sticky residue with a little distilled white vinegar (warmed slightly), rubbing alcohol, or a grease-cutting dish soap, then gently rub with an old toothbrush. These break down the gum base so the last traces lift out of the weave.
Wash the garment normally and let it air dry. Check that the gum is completely gone before machine drying, since the dryer's heat can melt any remaining bit deeper into the fabric and make it much harder to remove.