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We develop a theoretical framework to quantify how active forces renormalize the effective bending rigidity, Gaussian modulus, and surface tension of thermally fluctuating membranes. Building on classical statistical mechanics, we extend the analysis to include nonequilibrium active forces, both direct forces and those coupled to membrane curvature, within a nonlinear continuum formulation. Our model also incorporates hydrodynamic interactions mediated by the surrounding viscous fluid, which significantly alter the fluctuation spectrum. We find that direct active forces enhance long-wavelength undulations, leading to a substantial reduction in both the effective bending rigidity and surface tension, with the extent of softening strongly modulated by fluid viscosity. In contrast, curvature-coupled active forces primarily influence intermediate and short-wavelength fluctuations and show minimal sensitivity to viscosity. Together, these findings provide key insights into the nonequilibrium mechanics of active membranes and yield testable predictions for interpreting fluctuation spectra in both biological contexts and engineered membrane systems.