Regardless if it is or it is not an April fools joke it is a good opportunity to explain some points over this or similar situations.
You were born in Argentina
When you are two years old your parents decide to emigrate to, say Iceland
To leave Argentina your parents need to have your passport and DNI issued by the federal police, if not, you cannot leave the country.
Now you are 22 years old and decide to have a holiday to enjoy yourself and to know the country you born in
You have a passport issued by Iceland, and you travel on that passport to Argentina, when you arrive migrations tell you that you have so many days on that passport and then you need to leave the country (they know you were born in Argentina because the Iceland passport says so).
Now for some unforeseen reason or because you are plain stupid you pass the deadline and overstay, from that point you are not under the umbrella of Iceland protection, now you are under Argentinean law (actually since you put your feet in the country you are under Argentinean law because your status is an Argentinean native, no much any country can do about it, but in the other case if you did not overstay your visitors visa you did not broke any lawn then you can leave the country with your Iceland passport, now if you do break the law by over staying, then you will be detain to explain, if you are carrying two passports a Arg. and Ice. passports (both must by current) on you when you arrived to Argentina, then there is no problem, just present your Argentine passport to migrations to depart to Iceland no questions asked, but you cannot use your Iceland passport to leave because your stay period is overdue and it is an infraction, here Iceland cannot voucher for you because you are a Argentinean born citizen and you are of limits to the Iceland consulate, not options there.
If you born in Argentina and you are in Argentina regardless the passport you presented when entering the country you are under their law, also your dual citizenship status it is not recognized by Argentinean authorities, in Argentina you still Argentinean, be aware.
The only way I know is by renouncing to your argentine born status, but that is another story.