  The study of the Cosmic Microwave Background (CMB) lensing potential has
established itself by now as a robust way of probing the physics of large-scale
structure growth. The most common estimators of the lensing potential are
derived under the assumption of Gaussianity of the matter distribution and in
the Born approximation of the photon diffusion. In this paper we study the
performance of quadratic estimators when applied to realistic sky maps
extracted from multiple-lens ray tracing techniques in cosmological $N$-body
simulations. These are expected to model accurately the effects due to the
non-Gaussianity of the matter distribution induced by its nonlinearity and the
deviation from the Born approximation. We show that both these effects on their
own lead to reconstruction biases, but these tend to partially cancel each
other when both these effects are considered together. We forecast the impact
of these biases on the estimation of cosmological parameters for future
high-sensitivity CMB experiments like CMB-S4. We find that the cold dark matter
density, $\Omega_\textrm{cdm}$, the optical depth to reionization $\tau$, the
amplitude of primordial inflationary perturbations, $A_s$ and the sum of
neutrino masses, $M_\nu$, could be biased at the 1-2$\sigma$ level, if no
external data set is used. We also observe a reduction of the bias if external
data like baryon acoustic oscillations (BAO) is included.
