We study a physical model for the interaction between general inclusions bound to fluid membranes that possess finite tension, as well as the usual bending rigidity. We are motivated by an interest in proteins bound to cell membranes that apply forces to these membranes, due to either entropic or direct chemical interactions. We find an exact analytic solution for the repulsive interaction between two similar circularly symmetric inclusions. This repulsion extends over length scales of order tens of nanometers, and contrasts with the membrane-mediated contact attraction for similar inclusions on tensionless membranes. For non circularly symmetric inclusions we study the small, algebraically long-ranged, attractive contribution to the force that arises. We discuss the relevance of our results to biological phenomena, such as the budding of caveolae from cell membranes and the striations that are observed on their coats.