We consider a possibility to realize self-accelerating motion of interacting states with effective positive and negative masses in the form of pairs of solitons in two-component BEC loaded in an optical-lattice (OL) potential. A crucial role is played by the fact that gap solitons may feature a negative dynamical mass, keeping their mobility in the OL. First, the respective system of coupled Gross-Pitaevskii equations (GPE) is reduced to a system of equations for envelopes of the lattice wave functions. Two generic dynamical regimes are revealed by simulations of the reduced system, viz., shuttle oscillations of pairs of solitons with positive and negative masses, and splitting of the pair. The coaccelerating motion of the interacting solitons, which keeps constant separation between them, occurs at the boundary between the shuttle motion and splitting. The position of the coacceleration regime in the system's parameter space can be adjusted with the help of an additional gravity potential, which induces its own acceleration, that may offset the relative acceleration of the two solitons, while gravity masses of both solitons remain positive. The numerical findings are accurately reproduced by a variational approximation. Collisions between shuttling or coaccelerating soliton pairs do not alter the character of the dynamical regime. Finally, regimes of the shuttle motion, coacceleration, and splitting are corroborated by simulations of the original GPE system, with the explicitly present OL potential.