Throw away/in/out for rubbish?

Use "throw away" if you don't know where it's going to be thrown. "Throw out" is also common, though not as much. "Get rid of" is the most common, for if you're willing to throw out the idea of "throw".

"Throw in" always means to give up - cf. the common expression "throw in the towel". Don't use it to mean "throw away", unless you want to refer to a specific container to throw the thing in, in which case you would say "don't throw it in the garbage bin".

Throw away or throw out are pretty much synonymous, meaning 'get rid of'; either would do here. You could also say Don't bin them to emphasize the possibility of getting rid of them usefully. But throw in means 'give up', coming either from boxing where the second throws the towel into the ring to give up the fight, or from card-playing, as when a poker-player throws his cards into the middle of the table to indicate that he folds (there may be other roots, but the meaning is always similar)

The phrases throw out, throw away, and throw in the bin all carry the meaning to put in the trash. Therefore if you told me don't throw that away, I might still give it away to charity. You might instead want something like:

Please don't get rid of that.

Please hang on to that (for me)

These indicate that you want the person to maintain possession of the item, not just to avoid trashing it.

I grew up in the South and when someone told you to throw something out it was always pertaining to food that was to be "thrown out" the back door so the critters could eat it. If it was trash and meant to be tossed into the trashcan it was to be "thrown away." When a friend of mine came down from Buffalo he handed me a piece of trash and told me to throw it out and I was understandably concerned about his instructions.