Goo-gone makes me nervous, because it as a solvent it can itself damage some cases and coatings. You might try this: Remove the DVD(s), then pop the case in the freezer for a couple of hours. A lot of adhesives lose their stickiness at low temperatures and are very easy to remove. I do this all the time with plastic kitchenware and the labels come right off, usually in one piece.
This is less likely to damage a case than heat and/or moisture, and you don't have to be nearly as careful or sure-handed as with other methods.
Regards,
Joe "Ten Thumbs" DeMartino
