Absolutely no way. And another vote for hiding the passport, birth certificate and anything else like that.
However I would tell her that one day she could take him to visit south america, once he is old enough to be away from his mum and to look after himself - so she can't complain that you are being unreasonable. The fact that he's not going to be old enough for you to agree to this until he's at least 32 1/2 in your eyes is completely by the by - no meed to mention that trivial detail at this point. 
Whenever she asks, just keep repeating that he will be able to go at some point in the future but not until he's old enough...
Are there cultural reasons why the MIL would think that she is head of the household and should have sole control of a gc for such a long period of time? or is it just her personality? Did your dh spend long periods of time with other people when he was very young and therefore think it is normal? is she going to SA anyway and wants to take him with her to show him off before you get a chance to, so she gets all the credit so to speak for him? or is she specifically going to take your ds? Sorry, lots of questions!
Do you think your dh or mil would try to get a passport for him anyway for their home country - using a certified copy of the birth certificate and going to the embassy or doing it when in SA (if she is there anyway?) And if so, is there any way of a) finding out if it happens, b) stopping it from happening and c) if it does happen, getting the other passport revoked? Is there any way that you can put a mark on his passport at the moment to say that you don't agree to him being taken out of the country at the moment, in case your mil or dh decide just to do it when you are not around?
Is the country one of those that has signed up to that agreement that if one parent removes a child against the wish of the other parent, then it will return it to the other parent if there is a court order in place in the country where they are all living. Or once there will its laws and customs take precedence - ie if the child is with the father then they will say the child is with the right parent so tough to the mother.
Sorry for so many questions - hope you manage to persuade your dh and mil about how unreasonable they are being.
Just out of interest - you've said that you would also have a problem with your mum taking ds away for so long, so it's not about MIL taking him but about anybody taking him (even dh I am assuming) and him being away from you for so long when he is so little). What would he say if your mum said she would like to take him for 6 weeks somewhere - would he agree or thing she was crazy and absolutely no way?
Is your dh planning on going on this trip with MIL? Do you think that he is getting a bit tired of the constant demands of a toddler and is just seeing a chance for a selfish break from being a dad where he doesn't have to compete with a toddler but can just think about doing what he wants when he wants and therefore is able to cut himself off from the dad side of things because he sees some benefits for himself?
Definitely think you need to be happy to be on the attack for this one when talking about it with dh - he's told you that you're unreasonable. Remember that just because he's said it, it doesn't make it true. He's just hoping that the more he tells you that, you will come to believe it. You need to be telling him that he's the unreasonable one (along with MIL) and that it isn't going to happen. good luck!