I'm renting a car in Barcelona. And i would like to make a trip at the Pyrenees, both the Spanish and French sides. Are there any special requirements needed for the rented car?
You have to confirm with your rental agency that you're allowed to do that. In Europe most agencies allow you to travel in all countries of central and western Europe, but forbid you to visit countries in the East, like for example Belarus, Poland, Croatia, etc.
However, it will be likely that your rental agency will bill you an extra fee if you want to cross the border. Normally it is around 100 Euros.
Then a personal hint from me which you have to be careful about: There is no real border between Spain and France, so the chances are high that you're rental agency will not notice if you drive to France. However be aware if you have an accident in France and didn't tell the agency that you will cross the border, you will have big trouble.
Btw if we are talking about the Pyrenees: When planning your trip you should keep in mind that the roads there are really hilly and you will need much time to get from one place to another, even if it looks close on the map. And there are only a few passes between France and Spain which you can pass by car.
However I can really recommend the region. Was there this summer with my own car, and it is really nice there and worth the trip!
If your vehicle has a European style licence plate (i.e. EU flag on the left hand side and country code, e.g. E for Spain), you can legally drive all over EU. Some countries (Ireland) always have an EU style plate, some countries (UK) do not always have this style.
Car insurance should cover you all over the EU (normal car insurance anyway, I don't know about hire cars)
Ask/Tell your car hire company.
The Pyreenes are very nice. Try to take some scenic roads, rather than the main motorway.
I have no administrative advices, but just some suggestions for great sightseeings along the road: