We model theoretically the effect of localized forces on a fluid membrane anchored to a uniform elastic medium. We use this as a simple model for the plasma membrane of a cell. The atomic force microscope (AFM) has been used to apply such forces, but large membrane perturbations occurring in vivo are also treated within the same framework. Inclusions of this nature may include cell junctions, filipodia, caveolae, and similar membrane invaginations. The breakdown of linear elastic response, as observed by AFM, is predicted to occur for forces as small as 10 pN. We estimate the position of this crossover and the subsequent nonlinear behavior and make encouraging quantitative comparison with experiments. Intrinsic membrane inclusions interact through their overlapping strain fields. For similar, point force-like inclusions at large separations, this yields an attractive potential that scales like the inverse of their separation. For membranes that are intrinsically stiff or under tension, the binding force between inclusions can depend on the properties of the membrane and may be large enough to induce aggregation of inclusions, as observed experimentally. For inclusions that fix the magnitude of the membrane deformation, rather than the applied force, we demonstrate the possibility of metastable states, corresponding to finite separations. Finally, we discuss briefly the case in which inclusions couple to the membrane in more complex ways, such as via a torque (twist). In such cases, the interaction scales like a higher power of the separation, depends on the orientation of the inclusions, and can have either sign.