Is there a condition on vector fields $X$ and/or the underlying manifold $(M,g)$ such that $\nabla_X \omega$ is a closed differential form? That is, $d \nabla_X \omega = 0$. Let us assume for now that $\omega = df$, but if there is a more situations where a result holds then I would also be curious to know. Similar to this kind of question
When is the Hessian contracted with a vector field a closed form?
