I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $N\times S^1$.
If $f: M \to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^*d\theta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,\omega)$ is a manifold equipped with a nowhere vanishing closed 1-form, then in fact there is a fibration over the circle $f: M \to S^1$ so that $f^*d\theta$ is arbitrarily close to $\omega$. This is a theorem of Tischler, whose short proof can be read here.