Suppose $M$ is a smooth manifold and $X,Y\in\Xi(M)$ are two vector fields. Is it true that
If $\gamma(t)$ is a curve such that $X|_{\gamma(t)}=0$ then $[X,Y]|_{\gamma(t)}=0$
By now I have the feeling that it is false but I don't know how to prove it in a formal way.
It is false. Consider for example $X = y \frac{\partial}{\partial x}$ and $Y = \frac{\partial}{\partial y}$ on $\mathbb{R}^2$. Then
$$ X(Yf) - Y(Xf) = X \left( \frac{\partial f}{\partial y} \right) - Y \left( y \frac{\partial f}{\partial x} \right) = y \frac{\partial ^2 f}{\partial x \partial y} - \left( \frac{\partial f}{\partial x} + y\frac{\partial ^2 f}{\partial y \partial x}\right) = -\frac{\partial f}{\partial x} $$
and so $[X,Y] = -\frac{\partial}{\partial x}$ which is a non-vanishing vector field. If you take $\gamma(t) \equiv (0,0)$ then $X(\gamma(t)) = X((0,0)) = 0$ but $[X,Y]|_{(0,0)} \neq 0$.