We analyze a Bose-Einstein condensate (BEC) mixed with a superfluid two-component Fermi gas in the whole BCS-BEC crossover. Using a quasiparticle random-phase approximation combined with Beliaev theory to describe the Fermi superfluid and the BEC, respectively, we show that the single-particle and collective excitations of the Fermi gas give rise to an induced interaction between the bosons, which varies strongly with momentum and frequency. It diverges at the sound mode of the Fermi superfluid, resulting in a sharp avoided crossing feature and a corresponding sign change of the interaction energy shift in the excitation spectrum of the BEC. In addition, the excitation of quasiparticles in the Fermi superfluid leads to damping of the excitations in the BEC. Besides studying induced interactions themselves, we can use these prominent effects to systematically probe the strongly interacting Fermi gas.