They have put all the anti-holomorphic dependence into $\bar{v}(\bar{z})$. So for holomorphic modes of stress energy tensor $[L_n, \bar{v}(\bar{z})]=0$. For anti-holomorphic generators $[\bar{L}_n, \bar{v}(\bar{z})]\neq0$.
Field $X(z, \bar{z})$ have holomorphic and anti-holomorphic parts in Laurent expansion (2.89). Then from equation (2.40) with $h=\bar{h}=0$ you can find commutation relations for $L_n$ and anti-holomorphic modes of $X(z,\bar{z})$. It's straightforward to show, that result is zero.