I know that there are some restrictions on night time Eiffel Tower photos, but what exactly is allowed and what is not? Does it make a difference if some persons are on the photo and the tower is only in background? Are there also restrictions on day time photos?
Regarding the eiffel tower:
Image rights / the Eiffel Tower brand
By day, the sun light and the eiffel tower are in public domain, you can take it in picture and publish / try to sell the pictures as you want.
By night, the eiffel tower artistic lighting is under copyright if it's the main subject of the picture.
I think if it's for your little blog among your friends / a little audience, it should be ok (writing the "copyright Tour Eiffel - illuminations Pierre Bideau" is a plus).
If you have a big blog or try to make a commercial use, you must ask (and pay) for the right at the Eiffel Tower company (SETE).
Regarding someone in a picture:
If someone makes a picture of me, who owns the right to it? - photo.SE
Personality rights - Wikipedia
If the person is the main subject and you can recognize them, you have to ask them before to publish.
If they are far away and barely recognizable it's ok.
(There is an exception for public figure during public life.)
@travelot clearly laid out the most relevant rules but I would like to clarify something and highlight an important distinction: There is – to my knowledge – no general restriction on taking pictures of the Eiffel tower (the police might ask you to stop taking pictures, e.g. if you are disturbing traffic and the like but that's something else).
The issues with copyright or personality rights only come up if you want to publish these pictures in France. To the extent that copyright protection really applies, it does not in any case have any impact on your right to take pictures for your own private use. Publication in another country will be regulated by the local laws and international agreements (I don't really know but I doubt personality rights can generally be enforced abroad, copyright however is covered by international treaties but the details are complex).
Also, would-be right-holders (museums, the SETE, etc.) are generally not to be trusted for information on copyright, they routinely try to claim rights they don't have (e.g. on pictures of old artifacts stored in a museum) and to intimidate people who want to take pictures without any sound legal basis.
Travelot is arguably wrong. If not, there wouldn't be over 300 free photos of the Eiffel Tower here: https://commons.wikimedia.org/wiki/Category:Eiffel_Tower_at_night
The copyright owners would long since have formally demanded that wikipedia take them down, and they'd be long gone.
In terms of what tour-effel.fr says, Relaxed is right. Would-be right-holders are generally not to be trusted for information on copyright; they routinely try to claim rights they don't have, and to intimidate people who want to take pictures without any sound legal basis.
It may be that the legal system is corrupt enough for a nonsense decision in a local court to decide otherwise under financial or nationalist pressure, but would a US or other non-French court agree or enforce such a finding? Very doubtful.