Models for fluid deformable surfaces provide valid theories to describe the dynamics of thin fluidic sheets of soft materials. To use such models in morphogenesis and development requires to incorporate active forces. We consider active geometric forces that respond to mean curvature gradients. Due to these forces, perturbations in shape can induce tangential flows, which can enhance the perturbation leading to shape instabilities. We numerically explore these shape instabilities and analyze the resulting dynamics of closed surfaces with constant enclosed volume. The numerical approach considers surface finite elements and a semi-implicit time stepping scheme and shows convergence properties, similar to those proven to be optimal for Stokes flow on stationary surfaces.