Lattices of plasmonic nanorings with particular geometries exhibit singular, tunable resonance features in the infrared. This work examined effects of nanoring inner radius, wall thickness, and lattice constant on the spectral response of single nanorings and in Fano resonant square lattices, combining use of the discrete and coupled dipole approximations. Increasing nanoring inner radius red-shifted and broadened the localized surface plasmon resonance (LSPR), while wall thickness modulated the LSPR wavelength and decreased absorption relative to scattering. The square lattice constant was tuned to observe diffractively-coupled lattice resonances, which increased resonant extinction 4.3-fold over the single-ring LSPR through Fano resonance. Refractive index sensitivities of 760 and 1075 nm RIU(-1) were computed for the plasmon and lattice resonances of an optimized nanoring lattice. Sensitivity of an optimal nanoring lattice to a local change in dielectric, useful for sensing applications, was 4 to 5 times higher than for isolated nanorings or non-coupling arrays. This was attributable to the Fano line-shape in far-field diffractive coupling with near-field LSPR.