The key to using Goo Gone is not to use too much. Just a little with a few wipes will eradicate general crud. A more focused application is usually required for stickers as some are astonishingly tough to remove. While some due to age might simply fall off. Scotch tape, especially old scotch tape, demands elbow grease and nubile fingertips, but after removed a spray of this solution will make removal of sticky residue much easier.
Another reason to not use much is Goo Gone's tendency to lightly "cloud" certain types of plastics, like the dull Amaray clamshells from Trans World Entertainment releases. Glossier plastics like the clamshell pictured below are usually more resistant to this phenomena from heavier spray coats. I've found toilet paper to be the best rag for wiping not only your ass, but also the solution off tapes. Either spray the solution directly on the case gingerly or on the TP.
Of course, be extremely careful if using Goo Gone on the actual tape to clean or remove those annoying stickers. I can't imagine what it would do to magnetic tape, though I'd bet something bad. Also it's not a good idea to wantonly spray on actual paper covers, as it tends to soak into the paper. This can be done, some rental places were fucking stupid enough to sticker up the paper(?!), but be careful. Just spray some on a rag and lightly "tap" the sticker for the solution to soak into it. Wait a few minutes before attempting to peel it away. If the Goo Gone does soak into the cover paper, just let it sit out until it drys up. This rule goes along with common cardboard slip or big boxes.
Here's a ultra gunk encrusted tape I just cleaned up, Columbia Pictures Home Entertainment's 1983 VHS of The New Centurions. The cover paper is actually sealed within the clamshell so it's rare to find them in "restoreable" condition or not cut all up.
after (Billy Mays would be proud)