If, as you wrote, F(x, y, z)= 0 for all x, y, and z, it is constant no matter how x, y, and z are changed and all partial derivatives are 0. With x, y, and z functions of t, no matter how t changes, that simply results in x, y, and z changing so F remains constant. dF/dt= 0.
(This is NOT the question mathman answered. He appears to be thinking you were asking about the derivative being 0, not "f(u,v,w)=0".
As long as F(x, y, z) has continuous partial derivtives, and x, y, and z are differentiable functions of t,
\frac{dF}{dt}= \frac{\partial F}{\partial x}\frac{dx}{dt}+ \frac{\partial F}{\partial y}\frac{dy}{dt}+ \frac{\partial F}{\partial z}\frac{dz}{dt}
so that if all partial derivatives of F, with respect to x, y, and z, the dF/dt= 0 for any parameter, t. But if dF/dt= 0 for some t, it does NOT follow that the partial derivatives are 0.)
