Earthquake ruptures in poroelastic media involve a suite of complex phenomena arising from stick‐slip frictional instabilities and thermo‐hydromechanical couplings. In this study we propose a fully implicit, time‐adaptive, and monolithically coupled finite element model to simulate dynamic earthquake sequences in poroviscoelastic media. We consider a Kelvin‐Voigt viscoelastic material and characterize the impact of inertial effects on injection‐induced earthquakes. We present, for the first time, dynamic simulations of ruptures in rate‐and‐state faults in poroelastic media. Our simulations resolve the full earthquake cycle, including the interseismic, spontaneous earthquake nucleation, and dynamic rupture phases. We compare dynamic simulations with quasi‐dynamic ones, in which inertial effects are neglected and the slip singularity is resolved through a radiation damping approximation. Viscous dissipation models the physical process of seismic wave attenuation: As viscous damping increases, the patch size and the maximum fault slip become smaller, hence decreasing the expected earthquake magnitude. From a computational perspective, viscoelasticity helps avoid spurious high‐frequency oscillations during wave propagation. By including inertial effects, the dynamic model accounts for transient fluctuations of pressures and solid stresses during rupture, which are neglected in the quasi‐dynamic approach. Understanding these transient perturbations may shed light on the role of pore pressure in the mechanism of dynamic earthquake triggering. The poroviscoelastic dynamic approach is a good compromise between the inviscid, fully dynamic model, and the quasi‐dynamic one. A small amount of viscous damping allows us more efficient calculations, while preserving the most relevant features of dynamic ruptures, in particular slip velocities, accumulated slip, and seismic moment released.