We study a physical model for the interaction between general inclusions bound to fluid membranes that possess finite tension gamma, as well as the usual bending rigidity kappa. 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 approximately sqrt[kappa/gamma] and contrasts with the membrane-mediated contact attraction for similar inclusions on tensionless membranes. For noncircularly 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. These, and other, "gnarly buds" may prove fascinating to study further.