We analyze the magnon excitations in pyrochlore iridates with all-in-all-out (AIAO) antiferromagnetic order, focusing on their topological features. We identify the magnetic point group symmetries that protect the nodal-line band crossings and triple-point degeneracies that dominate the Berry curvature. We find three distinct regimes of magnon band topology, as a function of the ratio of Dzyaloshinskii-Moriya interaction to the antiferromagnetic exchange. We show how the thermal Hall response provides a unique probe of the topological magnon band structure in AIAO systems.