The evolution of a turbulent tangle of quantum vortices in the presence of finite-size active particles is studied by means of numerical simulations of the Gross-Pitaevskii equation. Particles are modeled as potentials depleting the superfluid and described with classical degrees of freedom following a Newtonian dynamics. It is shown that particles do not modify the building-up and the decay of the superfluid Kolmogorov turbulent regime. It is observed that almost the totality of particles remains trapped inside quantum vortices, although they are occasionally detached and recaptured. The statistics of this process is presented and discussed. The particle Lagrangian dynamics is also studied. At large timescales, the velocity spectrum of particles is reminiscent of a classical Lagrangian turbulent behavior. At timescales faster than the turnover time associated with the mean intervortex distance, the particle motion is dominated by oscillations due to the Magnus effect. For light particles, a nonclassical scaling of the spectrum arises. The particle velocity and acceleration probability distribution functions are then studied. The decorrelation time of the particle acceleration is found to be shorter than in classical fluids, and related to the Magnus force experienced by the trapped particles.