  The detection of a bright radio burst (hereafter FRB 200428) in association
with a hard X-ray burst from the Galactic magnetar SGR 1935+2154 suggests that
magnetars can make FRBs. We study possible neutrino emission from FRB-emitting
magnetars by developing a general theoretical framework. We consider three
different sites for proton acceleration and neutrino emission, i.e. within the
magnetosphere, in the current sheet region beyond the light cylinder, and in
relativistic shocks far away from the magnetosphere. Different cooling
processes for protons and pions are considered to calculate the neutrino
emission suppression factor within each scenario. We find that the flux of the
neutrino emission decreases with increasing radius from the magnetar due to the
decrease of the target photon number density. We calculate the neutrino flux
from FRB 200428 and its associated X-ray burst. The flux of the most optimistic
case invoking magnetospheric proton acceleration is still $\sim4$ orders of
magnitude below the IceCube sensitivity. We also estimate the diffuse neutrino
background from all FRB-emitting magnetars in the universe. The total neutrino
flux of magnetars during their FRB emission phases is a negligible fraction of
observed diffuse emission even under the most optimistic magnetospheric
scenario for neutrino emission. However, if one assumes that many more X-ray
bursts without FRB associations can also produce neutrinos with similar
mechanisms, magnetars can contribute up to $10^{-8} \ {\rm GeV \ s^{-1} \
sr^{-1} \ cm^{-2}}$ diffuse neutrino background flux in the GeV to multi-TeV
range.
