We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from $\mathbb Z^2$ via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight $λ$ of the non-planar edges is small enough, a suitably defined height function scales on large distances to the Gaussian Free Field with a $λ$-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, Kasteleyn's theory does not apply: the model is not integrable. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, which we then analyze via fermionic Renormalization Group methods.