  Lyman-$\alpha$ absorption systems at $z \sim 3$ with $N_{HI} \ga 3 \times
10^{14}$ cm$^{-2}$ have been found to be enriched with a mean metallicity of
$Z/Z_{\odot} \sim 10^{-2.5}$, and a large scatter in the metallicity. It is
reasonable to assume that the process of initial enrichment of the
intergalactic medium (IGM) at $z \ga 3$ also produced dust grains. We explore
the implications of the presence of dust grains in the IGM at high redshift, in
particular, the contribution of photoelectric emission from grains by hard
background photons to the net heating rate of the IGM.
  We show that (i) the charge on dust particles and the characteristics of
photo-emitted electrons differ substantially from those in the interstellar
medium (ISM) in several respects: a) grains are exposed to and charged by
photons beyond the Lyman limit, and b) because of this, the photoelectrons have
typical energy of tens of eV. We also show that ii) silicates are more
efficient heating agents than graphites; iii) small grains contribute mostly to
the net heating; iv) at densities typical of the IGM at $z\sim 3$ and for
Lyman-$\alpha$ absorbers, dust heating can be comparable or exceed
photoionization heating within an order of magnitude; v) this increases the
temperature of overdense regions, compared to the case of no dust heating, by
factor of $\sim 2$. We discuss the implications of this extra heating source in
Lyman-$\alpha$ absorbing systems.
