We evaluate the influence of the Coulomb drag of the electrons and holes in the gated n- and p-regions by the ballistic electrons and holes generated in the depleted i-region due to the interband tunneling on the current-voltage characteristics and impedance of the p+-p-i-n-n+ graphene tunneling transistor structures (GTTSs). The drag leads to a current amplification in the gated n- and p-regions and a positive feedback between the amplified dragged current and the injected tunneling current. A sufficiently strong drag can result in the negative real part of the GTTS impedance enabling the plasma instability and the self-excitation of the plasma oscillations in the terahertz (THz) frequency range. This effect might be used for the generation of the THz radiation.