What's the purpose of having a const function and mutable member variable? I see the consistency with RefCountClass, but I don't see why it's needed there either.

This is required so that const objects can still do reference counting. Otherwise we cannot pass around const objects, or destroy them.

Wouldn't it make more sense if those const objects use a mutable RefCountValue member variable?

The const_cast needs to be somewhere. I think it is better if it is buried here in this class and not in all code that interacts with this class.

Does the const_cast need to be anywhere if the RefCountValue is mutable?

struct RefCountValue
{
	int NumRefs = 0;

	void Release_Ref()
	{
		--NumRefs;
	}
};

struct StateMachine
{
	void Release_Ref() const
	{
		m_refCount.Release_Ref();
	}

	mutable RefCountValue m_refCount;
};

int main()
{
	const StateMachine machine;
	machine.Release_Ref();
}

Is this not a representative minimal example?

Yes, const cast happens when passing this pointer to delete function.

Right. Maybe you can add a comment for RefCountValue::Release_Ref that explains the use of const member function and const_cast on the object that'll be deleted.

Ok. Added a bunch of comments in EA/Westwood comment style.