The normal and the lateral Casimir interactions between corrugated ideal metallic plates in the presence of an amplifying or an absorptive dielectric slab is studied by the path-integral quantization technique. The effect of the amplifying slab, which is located between corrugated conductors, is to increase the normal and lateral Casimir interactions, while the presence of the absorptive slab diminishes the interactions. These effects are more pronounced if the thickness of the slab increases and, also, if the slab comes closer to one of the bounding conductors. When both bounding ideal conductors are flat, the normal Casimir force is nonmonotonic in the presence of the amplifying slab and the system has a stable mechanical equilibrium state, while the force is attractive and is weakened by intervening the absorptive dielectric slab in the cavity. Upon replacing one of the flat conductors with a flat ideal permeable plate the force becomes nonmonotonic and the system has an unstable mechanical equilibrium state in the presence of either an amplifying or an absorptive slab. When the left-side plate is a conductor and the right one is permeable, the force is nonmonotonic in the presence of a double-layer dissipative-amplifying dielectric slab with a stable mechanical equilibrium state, while it is purely repulsive in the presence of a double-layer amplifying-dissipative dielectric slab.