  Future ground-based gravitational wave (GW) detectors, i.e., Einstein
telescope (ET) and Cosmic Explorer (CE), are expected to detect a significant
number of lensed binary neutron star (BNS) mergers, which may provide a unique
tool to probe cosmology. In this paper, we investigate the detectability of the
optical/infrared electromagnetic (EM) counterparts (kilonovae/afterglows) from
these lensed BNS mergers by future GW detectors and EM telescopes using simple
kilonova, afterglow, and lens models. ET and CE are expected to detect
$\sim5.32^{+26.1}_{-5.10}$ and $67.3^{+332}_{-64.7}$ lensed BNS mergers per
year. We find that the EM counterparts associated with all these mergers will
be detectable by an all sky-survey in the H-band with the limiting magnitude
$m_{\textrm{lim}}\gtrsim27$, while the detectable fraction is $\lesssim0.4\%$
in the g-/z-band if with $m_{\textrm{lim}}\lesssim24$. Generally it is more
efficient to search the lensed EM counterparts by adopting the infrared bands
than the optical/UV bands with the same $m_{\textrm{lim}}$. Future telescopes
like Vera C. Rubin Observatory, China Space Station Telescope, and Euclid can
hardly detect the EM counterparts of even one lensed BNS merger. Roman Space
Telescope (RST) and James Webb Space Telescope (JWST) have the capability to
detect about a few or more such events per year. Moreover, the time delays and
separations between the lensed image pairs are typically in the ranges from
minutes to months and from $0.1$ to $1$\,arcsec, suggesting that both the GW
and EM images of most lensed BNS mergers can be well resolved by not only CE/ET
in the time domain but also RST/JWST spatially.
