You can definitely leave Argentina without the bike (as in you won't be stopped). Not sure about Bolivia or Paraguay and you definitely cannot do this from Chile.
However, if you do what Cleland says from Argentina, you will have a big headache if you ever come back - your passport will be marked with a bike that has overstayed and you will have a hard time explaining. Most likely you will be made to pay the duty on the bike as if you have imported it. And the duty will be calculated on some book value which will be a lot more than you think the bike is worth. And it will take a long time.
I would advise you to find a local Aduana office while in Argentina, and ask them if you can provably destroy the bike - in my experience they're quite helpful at answering hypothetical questions. All they're concerned about is preventing illegal importation.
I've heard rumours that it's possible to disengage your passport from the bike in Foz do Iguacu (which would enable you to just ditch it without consequences), but no idea of the details.
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