The infrared echo measurement distinguishes among vibrational line broadening processes according to their time scales. Rigorous computation of the echo signal for a realistic model requires quantum dynamical calculations for large anharmonic systems. The challenges posed by such calculations motivate the use of classical mechanical or semiclassical approximations. The application of such approaches to compute the vibrational echo signal depends on the nature and magnitude of quantum effects on the echo observable. We quantify these effects for a generic model of an anharmonic oscillator coupled to a harmonic solvent in the regimes of classical solvent and classical solute, classical solvent and quantum solute, and quantum solvent and quantum solute.