Interparticle forces that can be driven by applied (optical) fields could lead to the formation of new particle arrangements when assembled in arrays. Furthermore, the potentially large interactions and large local fields associated with plasmon excitations in anisotropic nanoparticles can lead to enhanced nonlinear responses and applications for sensing. These and other applications would benefit from simulations of spectra and forces arising from plasmonic interactions. We present the results of rigorous three-dimensional, finite-difference, time-domain calculations of near- and far-field properties of pairs of Au bipyramidal nanoparticles in three different configurations side-by-side, head-to-tail, and face-on. The absorption and scattering spectra depend strongly on the geometry as well as on the interparticle separation, as intuitively expected from a dipole coupling picture. Bipyramidal dimers in head-to-tail and face-on geometries exhibit an increasingly red-shifted (longitudinal) plasmon resonance with decreasing separation, whereas side-by-side dimers exhibit a blue shift. Large resonant field enhancements at the gap between particles in a head-to-tail configuration indicate the strong coupling of plasmonic modes. The Maxwell stress tensor formalism is employed to calculate the optical force one particle exerts on the other. Both significant attraction and weak repulsion can be obtained, depending on the relative arrangement of the particles. The force between bipyramids in the head-to-tail configuration can be greater than 10 times the force between pairs of Au nanospheres with the same volume. Experimental linear scattering spectra of particles trapped using the plasmon-resonance-based optical trapping method are found to be consistent with two particles trapped in the side-by-side configuration.