Travelling from Italy to Greece by Ferry with a French residency card but no passport

Article 21 of the Schengen Borders Code specifies that

The abolition of border control at internal borders shall not affect [...] (b) the possibility for a Member State to provide by law for an obligation to hold or carry papers and documents [...]

So, strictly speaking, you ought to check with Italy and Greece to see whether they require someone who is entering the country (by way of an internal Schengen border), who holds a residence permit from a Schengen country, to carry a passport in addition to the residence permit.

Also, if either country has a requirement that people generally carry identification, you should check whether your carte de séjour will satisfy that requirement. As an example, it seems that in the Schengen country I'm most familiar with, the Netherlands, it would not.

I looked very briefly at the web site of the Greek diplomatic mission to France (http://www.mfa.gr/france/fr/), and didn't see anything indicating one way or the other. Most information I found on the internet was geared towards travelers entering from outside the Schengen area.

In practice, it is probably sufficient to check with the ferry company. I checked the web site of http://www.greekferries.gr, and didn't find anything, but they do have a page advising that a passport or ID card is necessary to travel with an electronic ticket. Their customer service office ought to be able to tell you whether your carte de séjour will be acceptable as identification.

There are no official requirements that a transportation carrier must check any id documents for travel within the Schengen area. If a carrier requires you to present an id, it is to fulfil their own needs, e.g. to prevent black-market sales of personalized tickets or to authenticate print-at-home tickets. There are airlines allowing you to fly without having to present an id and there are train companies requiring you to do so.

As far as I can tell, the ferry route from Ancona to Patras is operated by both Anek Lines and Minoan Lines. Assuming that you are not an EU/EEA citizen, neither of these ferry companies allows you to travel without a passport.

Anek Lines requires you to enter the passport details to be able to book a ticket (from their General Terms and Conditions):

Passengers from non-European Union (and non-Schengen) countries are required to supply the following additional information: Passport number and expiry date, Visa expiry date (if required).

Minoan Lines require non-EU citizens to carry a passport during transportation (from their web page 'General Information'):

Passengers who are European Union citizens are not required to present a valid passport for Adriatic routes (Greece - Italy). However, please carry at all times with you an identification document. Passengers from other countries are required to carry a valid passport.