Tunneling of quasiparticles between two nearly aligned graphene sheets produces resonant current-voltage characteristics because of the quasiexact conservation of in-plane momentum. We claim that, in this regime, vertical transport in graphene/boron nitride/graphene heterostructures carries precious information on electron-electron interactions and the quasiparticle spectral function of the two-dimensional electron system in graphene. We present extensive microscopic calculations of the tunneling spectra with the inclusion of quasiparticle lifetime effects and elucidate the range of parameters (interlayer bias, temperature, twist angle, and gate voltage) under which electron-electron interaction physics emerges.