How can I check if I'm allowed to take a rental car from Argentina to Chile?

Consider using a small, not-so-well-known rental company. Big companies often have rules laid down by the legal/controlling/compliance department that branches have to follow, no exceptions allowed. One way that smaller competitors that lack the brand recognition can compete is through flexibility.

In November, I did a 2 week trip starting in Chile and dropping the car off in Argentina. With the big rental companies, that was either impossible or prohibitively expensive, so I chose a small local rental company and was very satisfied.

I have rented a car in the San Pedro de Atacama region. From there you can go to Bolivia and Argentina (though most people don't). As far as I remember I have consulted all major rental agencies in the area and almost all had an option in their booking process in order for you to be able to take the car out of the country at an additional cost of 100+ US$ or something like that. Although I think you would still have to return it to the same spot.

It may be more convenient to consult with smaller agencies as Michael Borgwardt suggested. They are generally a bit cheaper also.

You need to tell the hire company so you can get the necessary paperwork for the border controls. The car must go on a shared database which can take 48 hours to set up once you've signed the paperwork! Europe this is not.