We have a red Lego racing helmet that has become wedged with its round side up in a 2 handled Lego garbage can. I cannot access any of the visor indentations, the face slot or the neck circle. I tried banging on the upside down can, prying at the helmet with the little crowbar brick separater and broke a toothpick off between the helmet and the garbage can. I'm about to tell my kid that this can will just have to be considered full unless someone has a way to remove the helmet from it.
If the helmet were jammed head-down, you can use a needle-nose pliers to grasp the edge of the tube in the inside of the racing helmet, as shown, and then pull them apart. You may need to twist them apart as you pull.
In your case, since the helmet is jammed head-up, you need a way to push it out from the bottom. I suggest drilling a small hole through the tube recess in the bottom of the trash can. Unless you are intending to hold liquid in the trash can, there will be no functional impact whatsoever. The aesthetic impact is also minimal. I used a simple 3/32 wood drill bit I found in my garage. After drilling the hole, just push the drill bit through, and it should dislodge the helmet with minimal scratching. You can also pull the drill out and try pushing with another thin object. A wood skewer, for example, would be sufficiently rigid and less likely to scratch the helmet.
Short of damaging the parts (which oddTodd nicely documented, I'd tend to try to see if the part can't be removed by suction; although I'm not sure how you could generate enough.
Trying to inject a fluid underneath (with sufficient pressure) might do the trick too; that's what doctors do when dealing with clogged ears (usually with a sort of big syringe).
Lastly, you could glue some lever on the helmet and use it to pull it out, but you'd have to find a glue strong enough to make it work, and yet that you can remove afterwards. Note that if you value the trashcan more than the helmet (sounds unlikely, but you never know), this is the way to go (or you could use the drill trick with the helmet instead, I suppose).
Take a piece of hard sugar candy and screw a long thin screw into a small portion of it, small enough to fit in the garbage can without touching any of the sides. Wet the end of the candy to solve a bit of the sugar into a syrup and set it on the plastic helmet in the garbage can. Wait a day for it to dry, then pull the helmet out, and soak it in water to loosen the leftover candy on the helmet.
Thank you all for your answers. Now I must share the solution my 13 year old arrived at. I found two scratched elements and the pocketknife that pried them apart on my computer desk this morning. This was definitely NOT the best way to get a helmet out of a garbage can as both elements were damaged in the process, but the helmet (which now looks like it was in a high speed crash) is out and the can (not as scratched as the helmet) is empty.