My flight from Europe to South America includes 10 hours of transit time in New York (JFK). Is it possible to do a short sightseeing trip? Are there any city tours by bus that include the airport?
There are sightseeing buses but they hardly leave Manhattan, much less stop at JFK.
However, if you are very adventurous you could still do it. You won't have a lot of time, but can see a few of the most popular sights:
Either take the subway (metro) 1 hour for $8 to Manhattan, or a taxi half an hour (outside rush hours!) for $60. Go to Times Square, because why not. From there, you can easily get on a sightseeing bus, or just walk around on your own. Walk up Broadway to see Columbus Circle and Central Park, or down for the Empire State Building (only worthwhile if you will pay to ascend). If you like museums, the Museum of Modern Art is nearby and worth the steep admission price.
As an alternative: go downtown, e.g. Fulton Street. Here you can see Wall Street, the new One World Trade Center, take the free Staten Island ferry to see the Statue of Liberty (from afar, but not very far, 1 hour return). This is good if you like buildings more than people.
Forget Times Square!
Take public transport direct to SoHo. Spend, say, 3-5 hours being really cool and having fun. You need plan nothing - every single street, every corner, in the SoHo/Tribeca etc area is worth seeing. Take in a few cafes and maybe galleries. You can't go wrong.
Hop on public transport and go back for your flight. It's that easy.