We investigated the modulational instability (MI) of symmetric and asymmetric continuous-wave (CW) solutions in a model of a laser based on a dual-core nonlinear optical fiber. The model is based on a pair of linearly coupled cubic-quintic (CQ) complex Ginzburg-Landau (CGL) equations, that were recently shown to support several types of symmetric and asymmetric solitary pulses. We produce characteristics of the MI in the form of typical dependences of the instability growth rate (gain) on the perturbation frequency and system's parameters. In particular, the gain strongly depends on the spectral-filtering parameter and the CW amplitude itself. Generic outcomes of the nonlinear development of the MI are investigated by dint of direct simulations of the underlying equations. Three typical outcomes are found: a periodic chain of localized growing peaks; a stable array of stationary pulses (which is a new type of a stationary state in the model), and an apparently turbulent state.