Energy amplification in streamwise constant channel flows of viscoelastic fluids is studied from an input-output point of view by analyzing the responses of the velocity components to spatio-temporal body forces. These inputs into governing equations are assumed to be harmonic in spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number scaling of frequency responses from different input to different output components is developed. It is found that some of the components of frequency response peak at nonzero temporal frequencies. This is in contrast to the Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional timescales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio and show maximum for non-zero elasticity number. At low Reynolds numbers, the energy density decreases monotonically when the elasticity number is sufficiently small, but shows a maximum when the elasticity number becomes sufficiently large, suggesting that elasticity can amplify disturbances even when inertial effects are weak.