Along political lines, I would imagine you'd want to stay away from the Dirty War, Madres de la Plaza de Mayo, and so on. I have also heard that the Perons can be controversial as opinions about them are so polarized. And of course things like abortion, homosexuality, and so on.
On a personal level, I have found that Argentines are sensitive about their poor economy and the exchange rate (understandably so). It is considered extremely gauche to suggest that you are in their country because of this - instead you must only admit to visiting because of the wonderful culture, food, "buena onda" (good vibe) and so on. Also, they may be offended by any suggestion that non-Argentine products or destinations may be better than locally-made ones (cheese being one example). Argentina does not allow a lot of imports so there is less awareness about the products/strengths of other countries, plus a hefty amount of "buy local" and "travel local" propaganda issued by the government.
On the other hand, if you mention local favorites such as Fernet con Coca or mojellas (sweetbreads), you'll be an instant hit.