Let $Y$ be a closed oriented 3-manifold with Riemannian metric and suppose there exists a 1-form $v$ such that $dv=0$, $d\star v=0$ and |v|>0 in every point.
What will be the topology of this 3-manifold? Can we conclude that this 3-manifold is $S^1\times \Sigma$ with $\Sigma$ a Riemann surface?
If $M$ is fibered over $S^1$, you get a closed nonvanishing 1-form by taking pull-back of the angle form $d\theta$ from $S^1$. As for making this form harmonic, yes: See https://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric/75136.
Conversely, if $M$ admits a closed nonvanishing 1-form $\omega$ then $M$ fibers over $S^1$: This is a corollary of Tishler's theorem. Tishler's theorem requires $\omega$ to have integer periods but this is easy to arrange (the same argument as in my answer here).