We show that the extension of Casimir-like forces to fluctuating fluids driven out of equilibrium can exhibit two interrelated phenomena forbidden at equilibrium: self-forces can be induced on single asymmetric objects and the action-reaction principle between two objects can be violated. These effects originate in asymmetric restrictions imposed by the objects' boundaries on the fluid's fluctuations. They are not ruled out by the second law of thermodynamics since the fluid is in a nonequilibrium state. Considering a simple reaction-diffusion model for the fluid, we explicitly calculate the self-force induced on a deformed circle. We also show that the action-reaction principle does not apply for the internal Casimir forces between a circle and a plate. Their sum, instead of vanishing, provides the self-force on the circle-plate assembly.