This is a miniature model of a Black Hawk UH-60 helicopter (the chopper of choice for Team America) that was used to test different ways of shooting the movie. Bill Pope (Director of Photography for the film) wanted to see if he could use a photograph of a city laid out on a table and film that while holding the helicopter on a stick close to the camera to make 'in-flight' footage of the helicopter flying over Paris. To see if this would look believable to the audience, they made a couple of little grey models, put them on sticks and used a hand held mini video camera to experiment with. This is one of those helicopters. It also looks like they trying a couple of different fonts for the Team America logo.

Circled here is a little metal 'pipe' installed in the helicopter which is where a stick or metal rod would
be inserted to hold the helicopter during the test. It also has been cut open on the button and there is a
little red string attached which suggests they might have testing the scene where the Team
'rappels' out of the helicopter to attack the terrorist in Paris.

Here's a screen cap of Bill Pope conducting this test with a photo of Paris with a mini Eiffel Tower model.
In this image he is using a smaller helicopter mounted on a stick (not mine), but you can see what they were trying to do. The lower image is of the TV monitor connected to the camera. The last picture shows the same shot over Paris from the movie so you can see how it looked in the film. You can tell it's just a flat picture with a few 3-D models around the Eiffel Tower, but since your attention is focused on the helicopter, it goes mostly un-noticed.