Biological transport is supported by the collective dynamics of enzymatic molecules that are called motor proteins or molecular motors. Experiments suggest that motor proteins interact locally via short-range potentials. We investigate the fundamental role of these interactions by carrying out an analysis of a new class of totally asymmetric exclusion processes, in which interactions are accounted for in a thermodynamically consistent fashion. This allows us to explicitly connect microscopic features of motor proteins with their collective dynamic properties. A theoretical analysis that combines various mean-field calculations and computer simulations suggests that the dynamic properties of molecular motors strongly depend on the interactions, and that the correlations are stronger for interacting motor proteins. Surprisingly, it is found that there is an optimal strength of interactions (weak repulsion) that leads to a maximal particle flux. It is also argued that molecular motor transport is more sensitive to attractive interactions. Applications of these results for kinesin motor proteins are discussed.