Let $M$ be a Kahler manifold, with Kahler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $L_{X} g = 0$, where $L_{X}$ is the Lie derivative along $X$. Let $R$ be the Riemannian curvature tensor of $g$. Is $L_{X} R = 0$?

$g\mapsto R(g)$ is a natural (nonlinear) operation: it commutes with pullbacks by diffeomorphisms. This is just a transcription of: curvature transforms correctly under chart changes. Thus we get $L_X R(g) = dR(g)(L_X g)$ which implies "Yes"; two earlier comments also said this. 

