I'm a French citizen and I'm currently living in New York City with a 3 years work visa and would like to visit China for a week in October.
Can I apply for a tourism visa from the US (New York) without going back to France?
Yes, you can, according to the Consulate General of the People's Republic of China in New York (emphasis mine):
For entry into China, aliens shall apply for visas from Chinese diplomatic missions, consular posts or other resident agencies abroad authorized by the Ministry of Foreign Affairs, such as Chinese Embassies or Consulates. Each Embassy or consulate has its own consular districts. Usually, applicants are required to apply for visas from the relevant Embassy or Consulate. Applicants also can apply for visas at the nearest Embassy or Consulate.
Applicants who reside or travel away from the country of his or her nationality can apply for visas at a local Chinese Embassy or Consulate.
The New York City Consulate is located at 520 12th Avenue (between West 42nd and 43rd).