We show that arbitrarily weak interparticle interactions destroy the surface states of 3D topological superconductors with spin SU(2) invariance (symmetry class CI), in the presence of non-magnetic disorder. The interaction-driven instability results from the amplification of the density of states in local regions, due to wavefunction fluctuations induced by the disorder ("multifractality"). For 3D topological superconductors, the full spectrum of fluctuations can be computed exactly via 2D conformal field theory. We argue that time-reversal symmetry breaks spontaneously at the surface, so that topologically-protected states do not exist for class CI. The interaction-stabilized surface phase is expected to be an Anderson insulator exhibiting ferromagnetic order, or residing in an insulating plateau of the spin quantum Hall effect. We discuss the relationship between the CFT method employed here and the conventional (Finkel'stein) sigma model approach to disordered, interacting systems.