We investigate experimentally and theoretically the role of periodicity on the optical response of dolmen plasmonic arrays that exhibit a Fano line shape. Contrary to previous works on single nanostructures, this study deals with the in-plane near-field coupling between adjacent unit cells. By making an analogy to the electronic properties of atoms in the tight-binding model, specific behaviors of photonic states are investigated numerically as a function of the structural asymmetry for different coupling directions. These predictions are verified experimentally with dark-field measurements on nanostructure arrays which exhibit high tunability and fine control of their spectral features as a function of the lattice constants. These effects, originated from symmetry-breaking and selective excitation of the subradiant mode, provide additional degree of freedom for tuning the spectral response and can be used for the sensitive detection of local perturbations. This study provides a general understanding of the near-field interactions in Fano resonant lattices that can be used for the design of plasmonic nanostructures and planar metamaterials.