We develop a symmetry-controlled theory of anisotropic thermomagnonic torques in insulating altermagnets. Starting from the stochastic LLG equations, we identify a spin-splitter magnonic torque associated with thermally generated, sublattice-odd spin currents and an anisotropic entropic torque dictated by crystal symmetry. These torques produce anisotropic magnetic-texture responses to temperature gradients. In particular, thermally generated spin currents induce domain-wall precession, which reduces domain-wall velocities for selected gradient directions. We also predict an anisotropic skyrmion Hall response, with symmetry-selected directions enabling fast skyrmion motion with strongly suppressed transverse deflection. Our results reveal experimentally testable symmetry fingerprints of insulating altermagnets and extend more broadly to anisotropic magnets with exchange-driven magnon spin splitting.