Large-scale coronal waves associated with flares were first observed by the Solar and Heliospheric Observatory (SOHO) Extreme ultraviolet Imaging Telescope (EIT). We present the first three-dimensional MHD modeling of the interaction of the EIT waves with active regions and the possibility of destabilization of an active region by these waves. The active region is modeled by an initially force-free, bipolar magnetic configuration with gravitationally stratified density. We include finite thermal pressure and resistive dissipation in our model. The EIT wave is launched at the boundary of the region, as a short time velocity pulse that travels with the local fast magnetosonic speed toward the active region. We find that the EIT wave undergoes strong reflection and refraction, in agreement with observations, and induces transient currents in the active region. The resulting Lorentz force leads to the dynamic distortion of the magnetic field and to the generation of secondary waves. The resulting magnetic compression of the plasma induces flows, which are particularly strong in the current-carrying active region. We investigate the effect of the magnetic field configuration and find that the current-carrying active region is destabilized by the impact of the wave. Analysis of the three-dimensional interaction between EIT waves and active regions can serve as a diagnostic of the active region coronal magnetic structure and stability.