  Dense gas in minihalos with masses of $10^6-10^8~M_\odot$ can shield
themselves from reionization for $\sim100$ Myr after being exposed to the UV
background. These self-shielded systems, often unresolved in cosmological
simulations, can introduce strong absorption in quasar spectra. This paper is
the first systematic study on the impact of these systems on the Ly$\alpha$
forest. We first derive the HI column density profile of photoevaporating
minihalos by conducting 1D radiation-hydrodynamics simulations. We utilize
these results to estimate the Ly$\alpha$ opacity from minihalos in a
large-scale simulation that cannot resolve self-shielding. When the ionization
rate of the background radiation is $0.03\times10^{-12}~{\rm s}^{-1}$, as
expected near the end of reionization at $z\sim5.5$, we find that the incidence
rate of damped Ly$\alpha$ absorbers increases by a factor of $\sim2-4$ compared
to at $z=4.5$. The Ly$\alpha$ flux is, on average, suppressed by $\sim 3\%$ of
its mean due to minihalos. The absorption features enhance the 1D power
spectrum up to $\sim5\%$ at $k\sim0.1~h~{\rm Mpc}^{-1}~({\rm or}~10^{-3}~{\rm
km}^{-1}~{\rm s})$, which is comparable to the enhancement caused by
inhomogeneous reionization. The flux is particularly suppressed in the vicinity
of large halos along the line-of-sight direction at separations of up to
$10~h^{-1}~{\rm Mpc}$ at $r_\perp\lesssim2~h^{-1}~{\rm Mpc}$. However, these
effects become much smaller for higher ionizing rates
($\gtrsim0.3\times10^{-12}~{\rm s}^{-1}$) expected in the post-reionization
Universe. Our findings highlight the need to consider minihalo absorption when
interpreting the Ly$\alpha$ forest at $z\gtrsim5.5$. Moreover, the sensitivity
of these quantities to the ionizing background intensity can be exploited to
constrain the intensity itself.
