Most gun ranges will accept live ammunition for disposal in my area. I would check there, and if not there, a police department would be my next stop. This answer probably only works in the United States, I am not sure about abroad, and as a commentor pointed out, outside the United States, and especially in Europe, you may be finding military unexploded ordinance left over from the war. Don't try to pick up ordinance like unexploded artillery shells.
Additionally, shotgun shells and other ammunition often contain lead. If you are handling shotgun shells, expended or unexpended, be sure to wash your hands thoroughly before you eat, or wear gloves. Seperate the shells from other items in your pack so you don't contaminate anything. I never allow my children to handle expended shells because lead is particularly hard on children. In reality, picking up a few shells here and there probably isn't going to materially change your lead load, but best practices are certainly to be cautious and if you routinely encounter spent shells, make sure you have good lead abatement habits.
Shotgun shell specific: I have dealt with plastic hulled shotgun shells by using my pocket knife to score the outer shell so that I can remove the end. Then I dump out the pellets (remember they are likely lead), remove the wad, and then you are left with the two most dangerous parts of the shell, the powder and the primer. You can dispose of the powder in any number of ways. My favorite is to make a pile and light it on fire. Without containment, the powder just burns, it doesn't explode. You can also just scatter it and it won't be likely to pose a threat to anyone, however you haven't removed it from the environment at that point. As for the primer, don't try to do anything with it. It will explode if hit with a concussive force. A primer can kill a person if it is up close even without any sort of containment. Just take the leftover primer and shell to your local shooting range for disposal. Now you are hiking back much less AND if a shell happens to detonate in your pack (extremely unlikely), it is only the primer with no powder to ignite, so the damage will likely be nothing or close to nothing.