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://ru.wikipedia.org/wiki/%D0%98%D1%81%D0%BA%D1%80%D0%B8%D0%B2%D0%BB%D1%91%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5