Take a [warped product][1] $M=\mathbb S^1\times_f\mathbb S^1$ 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.

[1]:
https://arxiv.org/pdf/1307.0236.pdf