  The large total infrared (TIR) luminosities ($L_{\rm TIR} \gtrsim
10^{12}~L_\odot$) observed in $z \sim 6$ quasars are generally converted into
high star formation rates ($SFR \gtrsim 10^2~M_\odot$ yr$^{-1}$) of their host
galaxies. However, these estimates rely on the assumption that dust heating is
dominated by stellar radiation, neglecting the contribution from the central
Active Galactic Nuclei (AGN). We test the validity of this assumption by
combining cosmological hydrodynamic simulations with radiative transfer
calculations. We find that, when AGN radiation is included in the simulations,
the mass (luminosity)-weighted dust temperature in the host galaxies increases
from $T\approx 50$ K ($T \approx 70$ K) to $T\approx 80$ K ($T\approx 200$ K),
suggesting that AGN effectively heat the bulk of dust in the host galaxy. We
compute the AGN-host galaxy $SFR$ from the synthetic spectral energy
distribution by using standard $SFR - L_{\rm TIR}$ relations, and compare the
results with the "true" values in the simulations. We find that the $SFR$ is
overestimated by a factor of $\approx 3$ ($\gtrsim 10$) for AGN bolometric
luminosities of $L_{\rm bol} \approx 10^{12}~L_\odot$ ($\gtrsim 10^{13}~
L_\odot$), implying that the star formation rates of $z\sim 6$ quasars can be
overestimated by over an order of magnitude.
