In this work, we propose a well-balanced Implicit-Explicit Runge-Kutta scheme for the efficient simulation of the Baer-Nunziato model at all-Mach regimes. The numerical method is based on the explicit treatment of non-stiff terms, while stiff terms are kept implicit to remove the time step restriction stemming from acoustic waves. By introducing suitable linearizations in the semi-discrete energy equation obtained by a IMEX-RK time discretization, a predictor-corrector scheme is derived where the implicit contribution of pressure waves is accounted for by solving a system of non-linear elliptic equations in the unknown phasic pressures. A well-balanced discretization of non-conservative terms is also introduced, resulting in a numerical scheme capable of preserving steady-state solutions, including the so-called ”lake-at-rest” state. Finally, we prove that our numerical scheme is asymptotic-preserving. Compared to standard explicit finite volume schemes, the time step of our method is driven by the mean flow velocity and not by the speed of acoustic waves. Moreover, the proposed approach is characterized by a lower computational complexity if compared to fully (monolithic) implicit schemes. Numerical results demonstrate the capabilities of our numerical scheme to correctly capture shocks in the high Mach regime as well as to efficiently simulate flows at low Mach numbers.