We already know that looking for coins, even a simple 10 centimes Napoléon III, comes within the framework of the law on archeology and heritage since July 2016. This is indeed regulated, all finds that affect humanity comes back to the State.
For fossils, we know that it is strictly forbidden to search in a geological natural reserve. There is no specific law to paleontology or geology. A bill was introduced in 1997, but nothing is clear right now.
However, there is a law that can be taken into account, it is the article 552 of the Civil Code which concerns the owner of the soil and the objects found there. Everything above and below the ground belongs to the owner.
When the geological specimens are clamped in the basement, it is real estate (immovable). When they are on the floor and are removable, it is considered as a furniture (movable object).
If a person finds and takes a fossil, it can be considered a theft of movable property. Unless you have the agreement of the owner of the land.
Similarly, detaching fossil specimens from the basement can be considered as a deterioration of real estate.
See the concept of immovable and movable properties of the Civil Code.
You must therefore have the agreement of the owner of the land. If it is the state, you need the agreement of the town hall, and to ask to the prefecture if it is necessary to get other agreements.
So, if an embittered person sees you digging in the ground, or beating pickaxe, hammer and chisel in the rock, and that he calls the nationale gendarmerie, then unfortunately the law will be against you, unless you have the necessary agreements.