We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle and are related to symplectic geometry and Seiberg-Witten theory.
We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.
D. Kotschick. "On products of harmonic forms." Duke Math. J. 107 (3) 521 - 531, 15 April 2001. https://doi.org/10.1215/S0012-7094-01-10734-5