I am a French citizen living in the US for many years with a green card. I am looking to obtain my US citizenship. Will I loose my French citizenship or can I keep dual nationalities?
Generally speaking, French law does not make multiple citizenship difficult in any way. You don't need to renounce any prior nationality to become French and there are very few situations in which you lose it when you are. Perd-on la nationalité française en acquérant une autre nationalité ? is a brief overview of the latter aspect from the official vosdroits.service-public.fr website.
For a long time France was a party to the 1963 Strasbourg convention, which did provide for the loss of one's first citizenship when acquiring that of another party to the convention. But it had some exceptions and the US was never part of it (it was strictly a European thing). Since then, France denounced the relevant chapter of the Strasbourg convention so that all this is only of historical interest.
I don't really know about the US end of things but I assume it should be OK too as the US tends to be permissive in this respect and there are many dual US-something citizens.
According to Wikipedia on French nationality law:
Dual citizenship has been permitted since 1973. Possession of one or more other nationalities, does not, in principle, affect the French nationality.
(I'm afraid I don't have an official reference for that.) For the US side, acquiring US citizenship does not require that you give up other citizenships. I have a Dutch friend who obtained US citizenship by naturalization and continues to hold both citizenships. It can, however, get somewhat more complicated to comply with the taxation regulations of both countries of citizenship.
The answer is unequivocally: "NO". You can have as many citizenships as you wish as long as you comply with entry exit passport-rules of each respective country. Example: If you become US citizen you can no longer enter US as French national using French passport.
The US Immigration Oath that one must make at the time one is sworn in requires one to "renounce and abjure" all previous nationalities and titles of nobility.
Whether the government of your previous nationality would be informed of this action or recognises US naturalisation as an explicit renunciation would need to be cleared with them.