The FAA only has Jurisdiction over US airspace and you must follow the rules of the airspace you are flying in/through. If you are talking about N registered airplanes they must follow FAA rules for airworthiness etc. even if they are kept outside the country.
Here is in interesting question on breaking laws on a plane in general (sort of applicable here)
Side Note: Although the FAA does not have jurisdiction outside US airspace if there is an accident involving a US built aircraft (mainly Boeings at this point) the NTSB will send a representative to compile their own report (and aid in the investigation). I know this happens for commercial accidents I am unaware if they send representatives for US build GA plane accidents outside the country.
Side Note 2: US pilots licenses are valid elsewhere in the world and allow you to fly elsewhere in the world certain actions elsewhere in the world (if caught/reported) may lead to the revocation of your US issued license. There is some info on that here