Take a warped product $M=\mathbb S^1\times_f\mathbb S^1$ for a nonconstant smooth function $f$.
The horizontal vector field $V$ satisfies your condition, but for harmonic function is not invariant $f$, the function $Vf$ is not harmonic.
To see this, consider the $V$-flow $\Phi^t\colon M\to M$. If $Vf$ is always harmonic then so is $f\circ\Phi^t$ for any $t$. The latter fails evidently.