The only time it's disappointed me was when I was trying to use it -- among a variety of other substances -- to deal with something other than sticker goo.
A few years back, a bunch of electronic devices were sold in the US that used some sort of rubberized coating that gave them a nice grippy feeling. Unfortunately, it turns out that after a couple of years, the coating degraded and turned into an incredibly sticky mess. I had a Grundig G6 Aviator shortwave radio that was affected, as well as a few other devices. It felt a bit like sticker adhesive, but trying to clean the stuff off with Goo Gone didn't work well (ultimately, isopropyl alcohol and a lot of elbow grease wound up being my most-successful combination).
But for what it is actually billed for, adhesive residue, I've had good experiences.
Note that Goo Gone, or at least the variant I have, has a quite-strong citrus smell, which I assume is there to mask some less-pleasant smell that the active ingredients have. So when I use the stuff, everything nearby smells like oranges for a while. Haven't had a situation where that's a problem yet, but thought I'd mention it in case it would be an issue for anyone else using it.
It really is amazing. Was able to scrape a LOT of gorilla tape (it's like FlexTape) residue off of a painted surface using a few applications of goo gone, and an improvised plastic squeegee last weekend. No scratches on the paint, and no more than about 2 minutes of effort on my part.
Or any kind of oil. Even WD-40, if you have it. Put oil on the adhesive residue, then scrub it off with a cloth. You can then remove the oily residue with soap.
Lots of good suggestions in this thread already, but long ago I used to work at a company that sold second-hand stuff where we often had to deal with stubborn labels and I haven't seen our "nuclear option" mentioned yet. Brake cleaner, which is basically an organic solvent in a spray can, should be great on glass. Be careful with it, though, it melts some kinds of plastics. And ensure it's used in a well-ventilated area.
Try that if the usual GooGone type stuff doesn't work.
We also used WD40, it tended to be gentler on plastics. Though it left a greasy residue that we'd need to clean off afterward and it was pretty slow.
I suggest, if it's not too gooey, grab some packing tape, and press it on the adhesive. Then, pull it off quickly. You might have to do this a few times, but it's the best method I've found that doesn't leave residue, and also works with most sticky adhesives.
I have had great luck removing labels by filling bottles / jars with boiling water and letting them sit a few minutes. I mostly do this to save wine labels I like or want to remember. When doing this it is also very easy to wipe off any residue.
Years ago at one job we used to use a combination of a razor scraper and Goof Off and it worked well. It was on pricing stickers on metal painted with enamel, and as long as you didn't dig into it and just worked it loose it would come clean. Goof Off is a harsher chemical than Goo Gone (I think toluene or some mixture) so not the best for long term exposure, but that was then and we had some ventilation. I also think the product sold now like so many might be more diluted and not the original.
Olive oil on the other hand works surprisingly well if it can get to the adhesive.
I like to use WD-40, I just lube it up and massage the adhesive until it dissolves into a goo that can be removed with dish soap. I find the goo-gone that I tried evaporates too quickly to soak into stiffer jar label adhesives.
i haven’t tried olive oil and other similar oils like others have suggested, but there’s a product in my country that’s specifically sticky stuff remover (like label residue) and it’s made of orange oil
It's in the peels and has lots of the fruits flavour. All citrus have it, that's why you grind a bit of lemon peel into your food: It tastes way more lemony than the juice.
Goo-B-Gone or Goo-Gone, plus steel wool like SOS pads or Scotch-Guard. Razors would work if it was a flat surface, but on a curved one like a bottle or glass, you want the flexibility of the steel wool.
Alcohol solvent can make this worse depending on the adhesive. It can just spread the stuff and thin it out. Someone else suggested oil, that's the ticket. Bit of olive oil on a rough cloth brings it right up. Wd40 was also suggested, same idea. You want it to move easily, not dissolve.
Nobody's mentioned lighter fluid so far, but that stuff works well. Dad always used Ronsonol and it's quite cheap. It's also highly flammable, so mind that.
The lighter fluid that the parent poster is talking about, Ronsonol, is not the stuff that goes in butane lighters, nor is it the "lighter fluid" used to start charcoal barbecues. He's talking about the stuff that goes into Zippo and similar refillable lighters, stuff that's also called "naphtha".
If you want to use it as a solvent with much frequency, you can get the stuff at dramatically-lower unit prices in much larger containers than what Ronsonol is sold at.
Keep in mind that this is potent stuff and can damage some surfaces. For example, another user is talking about using Goo Gone above to remove adhesive from paint on a wall. Naphtha is more-or-less paint thinner -- your wall's paint will dissolve in it. It'd be fine on glass, what OP is intending it for, but if you use it elsewhere, be careful with it, try it on a small amount of the surface somewhere first to see if it damages it.
It can also cause skin irritation. Probably not a big deal unless you're regularly using it, but worth keeping in mind.
I also have a container of white spirits, which are similar but less volatile.
An Oxyclean soak is the most common way I see homebrewers recommend for removing labels from bottles. Definitely a good choice if being food safe matters.