An approach based on effective field theory (EFT) is discussed and applied to the problem of surface-mediated interactions between rigid inclusions of circular footprint on a membrane. Instead of explicitly constraining the surface fluctuations in accord with the boundary conditions around the inclusions, the EFT formalism rewrites the theory; the Hamiltonian of a freely fluctuating surface is augmented by pointwise localized terms that capture the same constraints. This allows one to compute the interaction free energy as an asymptotic expansion in inverse separations in a systematic, efficient, and transparent way. Both entropic (fluctuation-induced, Casimir-like) and curvature-elastic (ground-state) forces are considered. Our findings include higher-order corrections to known asymptotic results, on both the pair and the multibody levels. We also show that the few previous attempts in the literature at predicting subleading orders missed some terms due to an uncontrolled point-particle approximation.