  Neutrino-electron scattering experiments can explore the potential presence
of a light gauge boson $A'$ which arises from an additional $U(1)_{B-L}$ group,
or a dark photon $A'$ which arises from a dark sector and has kinetic mixing
with the SM hypercharge gauge field. We generically call it a dark photon. In
this study, we investigate the effect of the dark photon on neutrino-electron
scattering $\nu e^-\rightarrow\nu e^-$ at the newly proposed forward physics
experiments such as FASER$\nu$, FASER$\nu2$, SND@LHC and FLArE(10 tons). We
estimate the anticipated sensitivities to the $U(1)_{B-L}$ gauge coupling in a
wide range of the dark photon mass $M_{A'}$. We compare the sensitivities of
the proposed forward physics experiments with the current limits from TEXONO,
GEMMA, BOREXINO, LSND, and CHARM II as well as NA64e experiments. We also
extend the calculation to obtain the sensitivities on the kinetic mixing
parameter $\epsilon$ in a wide range of dark photon mass $M_{A'}$. We
demonstrate that the sensitivities do not improve for $M_{A'} < 1 $ GeV at the
Forward Physics Facilities.
