  We use cosmological hydrodynamic zoom-in simulations and semi-analytical
models to study the effects of primordial black holes (PBHs) on first star
formation. Our models self-consistently combine two competing effects: initial
(isocurvature) perturbations induced by PBHs and BH accretion feedback.
Focusing on PBHs with masses $\sim 30\ \rm M_{\odot}$, we find that the
standard picture of first star formation in molecular-cooling minihaloes is not
changed by PBHs, as the simulated star-forming gas clouds in the central parsec
are very similar to those in the $\rm \Lambda CDM$ case when PBHs make up
$f_{\rm PBH}\sim 10^{-4}-0.1$ of dark matter. With a dynamical friction
timescale of $\sim 2-10\ \rm Myr$ when the central gas density reaches $10^{5}\
\rm cm^{-3}$, it is also unlikely that PBHs can sink into star-forming discs
and affect the evolution of protostars, although they may interact with the
stars during the main-sequence stage. At larger scales, PBHs tend to shift star
formation to more massive haloes, and accelerate structure formation. The
latter effect is stronger in regions with higher initial overdensities. For
$f_{\rm PBH}\sim 10^{-4}-0.01$ (allowed by observational constraints), the
collapsed mass fraction of haloes hosting Population III stars is similar
(within a factor of $\sim2$ at $z\lesssim 30$) to that in $\rm \Lambda CDM$,
implying that the impact of stellar-mass PBHs on the cosmic star formation
history at $z\gtrsim 10$ is small. We also find that the Lyman-Werner photons
from PBH accretion in atomic-cooling haloes may facilitate the formation of
direct-collapse BHs.
