Network structure has a large impact on constant-selection evolutionary dynamics, with which multiple types of fitness (i.e., strength) compete on the network. Here we study constant-selection dynamics on two-layer networks in which the fitness of a node in one layer affects that in the other layer, under birth-death processes and uniform initialization, which are commonly assumed. We show mathematically and numerically that two-layer networks are suppressors of selection, which means that they suppress the effects of the different fitness values among the different types on the final outcomes of the evolutionary dynamics (called fixation probability) relative to the constituent one-layer networks. In fact, many two-layer networks are suppressors of selection relative to the most basic baseline, the Moran process. This result is in stark contrast with the results for conventional one-layer networks for which most networks are amplifiers of selection.