Topological magnonic materials have attracted much interest because of the potential for dissipationless spintronic applications. Pyrochlore iridates are theoretically regarded as good candidates for designing topological magnon bands. However, experimental identification of topological magnon bands in pyrochlore iridates remains elusive. We explored this possibility in Y2Ir2O7 using Raman spectroscopy to measure both the single-magnon excitations and anomalous phonon shifts. From the single-magnon energies and tight-binding model calculations concerning the phonons, we determined the key parameters in the spin Hamiltonian. These confirm that Y2Ir2O7 hosts a nontrivial magnon band topology distinct from other pyrochlore iridate compounds. Our work demonstrates that pyrochlore iridates constitute a system in which the magnon band topology can be tailored and that Raman spectroscopy is a powerful technique to explore magnon band topology.