Legally speaking, you need a valid passport or equivalent travel document to cross any border of a Schengen state, whether it's an inner or outer border of the Schengen area. I can't find a list of equivalent travel documents for France, but the one for Italy only lists “exotic” alternatives — documents for refugees, for seafarers, etc. (Nationals of an EU country and a few others need only an identity card).
In practice, borders inside the Schengen area are not equipped for systematic checks and there is a high likelihood that you'll just drive through without even noticing that you've passed the border. There are (or used to be) occasional checks for drug smuggling on the road from the Netherlands to France; I doubt that it's a problem if all you can show is a driving license. It's better if you have your residence permit and a photocopy of your passport — with these documents, it would take exceptional to get into trouble.
As far as I know, there is no direct penalty for not carrying an identity document in France. (I know that for nationals; I think it's also true for non-nationals.) There is an obligation that you must be able to prove your identity upon request (technically, the request has to be justified, but in practice a police officer can always find a pretext). As a non-national, you'd also have to prove your right to be there, as otherwise you could be detained and expelled. Again, with a proof of identity such as a driving license, plus a copy of your passport and residence permit, you are very unlikely to get into trouble.
The one thing you cannot reliably do without a passport is fly. All commercial airlines require a passport (or identity card), even for Schengen internal flights. They often don't check, but if they do, they're likely to not let you board the flight if you don't have documentation that's acceptable to them.