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The robustness against local perturbations, as long as the symmetry of the system is preserved, is a distinctive feature of topological quantum states. Magnetic impurities and defects break time-reversal invariance and, consequently, time-reversal invariant (TRI) topological superconductors are fragile against this type of disorder. Non-magnetic impurities, however, preserve time-reversal symmetry and one naively expects a TRI topological superconductor to persist in the presence of non-magnetic impurities. In this work, we study the effect of non-magnetic disorder on a TRI topological superconductor with extended $s$-wave pairing, which can be engineered at the interface of an Fe-based superconductor and a strongly spin-orbit coupled Rashba layer. We model two different types of non-magnetic random disorder and analyze both the bulk density of states and edge state spectrum. Contrary to naive expectations, we find that the disorder strongly affects the topological phase by closing the energy gap, while trivial superconducting phases remain stable and fully gapped. The disorder phase diagram reveals a strong expansion of a nodal phase with increasing disorder. We further show the decay of the helical Majorana edge states in the topological phase and how they eventually disappear with increasing disorder. These results alter our understanding of effects of impurities and disorder on TRI topological phases and may help explain the difficulty of experimental observation of TRI topological superconductors.