If I'm a dual Polish/American citizen traveling to Argentina from the U.S., will I have to pay a visa fee if I use my Polish passport?

I don´t think so. The visa is for reciprocity reasons and Poland doesn´t require a that document to Argentine citizens so Im preety sure It will be Ok. However, be sure that the passport is up to date.

The new $160 fee is a reciprocity fee for US Citizens and only applies when traveling through one of the two Buenos Aires international airports. It must be paid online prior to arrival. You can see step-by-step instructions for payment at: http://www.argentinafees.com

Be careful, however, not to re-enter the US using your Polish passport.  The act of a dual citizen entering the US on another nation's passport has been used by US authorities to jeopardize the holder's US citizenship.  "Most U.S. citizens, including dual nationals, must use a U.S. passport to enter and leave the United States (http://www.travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html)"