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We propose an electrically driven spin injector into normal metals and semiconductors, which is based on a magnetic tunnel junction (MTJ) subjected to a microwave voltage. Efficient functioning of such an injector is provided by electrically induced magnetization precession in the "free" layer of MTJ, which generates the spin pumping into a metallic or semiconducting overlayer. We theoretically describe the spin and charge dynamics in the CoFeB/MgO/CoFeB/Au(GaAs) heterostructures. First, the magnedynamics in the free CoFeB layer is quantified with the account of a spin-transfer torque and a voltage-controlled magnetic anisotropy. By numerically solving the magnetodynamics equation, we determine dependences of the precession amplitude on the frequency $f$ and magnitude $V_\mathrm{max}$ of the ac voltage applied to the MTJ. It is found that the frequency dependence changes drastically above the threshold amplitude $V_\mathrm{max} \approx 200$mV, exhibiting a break at the resonance frequency $f_\mathrm{res}$ due to nonlinear effects. The results obtained for the magnetization dynamics are used to describe the spin injection and pumping into the Au and GaAs overlayers. Since the generated spin current creates additional charge current owing to the inverse spin Hall effect, we also calculate distribution of the electric potential in the thick Au overlayer. The calculations show that the arising transverse voltage becomes experimentally measurable at $f = f_\mathrm{res}$. Finally, we evaluate the spin accumulation in a long n$^+$-GaAs bar coupled to the MTJ and determine its temporal variation and spatial distribution along the bar. It is found that the spin accumulation under resonant excitation is large enough for experimental detection even at micrometer distances from the MTJ. This result demonstrates high efficiency of the described nanoscale spin injector.