My wife is in France with her own 1 year long stay visitor's visa. She will give birth to our daughter shortly. When the baby is born will it be subject to the 90 Schengen requirement like any other US Citizen visiting France and then be required to leave and apply for a French visa or can it remain in France (Schengen Zone) with it's mother for the remainder of her visitor's visa?
It looks to me like your child will need a titre d'identité républicain (TIR):
Le titre d'identité républicain (TIR) peut être délivré à un mineur né en France de parents étrangers titulaires d'un titre de séjour. Il lui permet de prouver son identité et d'être dispensé de visa lors de son retour en France après un voyage à l'étranger. La personne qui exerce l'autorité parentale sur l'enfant doit en faire la demande en préfecture ou en sous-préfecture en fournissant certains justificatifs. Le document est valable 5 ans et renouvelable.
Machine translation with human copy editing:
The Title of Republican Identity (TIR) can be issued to a minor born in France of foreign parents holding a residence permit. It allows the child to prove his or her identity and to be exempt from visa when returning to France after a trip abroad. The person who exercises parental authority over the child must make the request in the prefecture or sub-prefecture by providing certain supporting documents. The document is valid for 5 years and renewable.
The cost is EUR 45 (assuming neither you nor your child has EU, EEA, or Swiss citizenship).
Although this document is not a uniform-format residence permit, it is included in the list of residence permits issued by member states (pdf), which means that (after receiving the document, at least) your child's time spent in France will not count toward the 90-day limit in the Schengen area. This in turn means that your child will be able to visit other Schengen countries within the 90/180 day limit.
As @SJuan76 stated, your wife's one-year long-stay visa is the equivalent of a titre de séjour or residence permit in France, as long as she had it validated at the OFII (or Office français de l'immigration et l'intégration). Technically she would have had to do this if the visa is valid for one year.
Thus, with that and the birth certificate (extrait d'acte de naissance) of your child, and the other required documents (proof of address, etc.) as stated in @phoog's link.