It's completely normal for a puppy, especially a mouthy breed like a Lab, to pick up everything and even eat it.
When I get my dogs as pups, I never play keep-away with them (when they are the ones that have the thing). They can play that with their dog friends, that's fine, but we don't play it. I teach them to bring me stuff and we trade. I always keep treats on me for this reason.
You can start at home with toys, high value stuff, etc. You can give him something good, trade for something great, then give him the good thing back. For example, if he has a bully stick you might trade for a piece of cheese if he likes that better, let him eat it then give him the bully stick back. It becomes a game.
Same goes for drop it or leave it. You trade for a treat, but start with items where it's not an emergency, unlike glass or a bone. If he has a toy, you can ask for a drop it, then treat and praise. You can hold a toy and when he goes for it, ask him to leave it, while putting the treat under his nose (making it really obvious) and give him the treat and praise him for leaving it.
In public, you should watch him closely and tell him to leave it before he gets the thing he wants and then give him a treat. You need to try to stay ahead of the game, but be prepared in the event the pup stumbles across something tasty that you didn't notice. Keeping him in a heel position that is not too far away is also helpful.
I've found that keeping it fun and a game is key. Otherwise, dogs tend to play keep away, hide and gobble, run and gobble or even get defensive and run under a bed or table and growl, even bite. When people don't train their dogs and instead are angrily running at them and pulling things from their mouths or have them on the floor prying their mouths open and freaking the dog out, you typically run into trouble. You want to avoid that.
Linus learned leave it and drop it very quickly when he was three months old. We went through probably three lbs of Charlie Bears at a beach trip. Every five seconds I was asking him to leave or drop the shell, crab, dead seagull, seaweed, rotting fish and the list goes on. By the end of the trip he was picking stuff up, running to me, then dropping it for a treat. It's a phase that puppies will go through while learning, so I let it play its course, and now they are solid, even off leash and at a distance.
This is pretty common and I'm sure it is covered in a book with better instructions or in your obedience class. Ian Dunbar has some good pet training books.