A one-dimensional diatomic chain with harmonic intersite potential and nonlinear external potential is considered (the Klein-Gordon model). Localized solutions of the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum-"gap breathers"-are studied numerically. The linear stability analysis for these solutions is performed while changing the system parameters from the anticontinuous to the continuous limit. Two different types of solutions are considered: symmetric centered at a heavy atom and antisymmetric centered at a light atom, respectively. Different mechanisms of instability, oscillatory as well as nonoscillatory, of the gap breathers are studied, and the influence of the instabilities on the breather solutions is investigated in the dynamics simulations. In particular, the presence of an "inversion of stability" regime, with simultaneous nonoscillatory instabilities of symmetric and antisymmetric solutions with respect to antisymmetric perturbations, is found, yielding practically radiationless mobility.