Paris’ Eiffel Tower is part of the public domain, and as such your day-time photos are a-ok. But night-time photos are different story. The building is illuminated at night with various light shows, and those are ruled as copyrighted works of art, meaning “reproducing” it through photos, and sharing those photos online, requires permission.
You can think of it as similar to laws that make it illegal to record a stage play or movie while in the theater, just taken extremely literally. And this isn’t just a French thing either, as Torrent Freak notes there are several other countries with copyrights on architectural structures. Even the website for the Société d’Exploitation de la Tour Eiffel says that commercial use of the Eiffel Tower, which includes sharing photos online, requires permission.
This issue was brought to light recently by the EU Observer, who point out that Belgium and Italy have copyrights on public buildings. One famous Belgian landmark, for example, is the Atomium. As a result, Wikipedia entries feature censored images (such as the Norwegian page), or photos of models of the building (like the English page).
So it appears that posting your favorite shot of the Eiffel Tower just after sunset on Facebook is technically a copyright infringement, you probably have little to fear in terms of enforcement, judging by the significant number of night-time pictures on Instagram and Google image search.