  Couplings between light scalar dark matter (DM) and neutrinos induce a
perturbation to the neutrino mass matrix. If the DM oscillation period is
smaller than ten minutes (or equivalently, the DM particle is heavier than
$0.69\times10^{-17}$ eV), the fast-averaging over an oscillation cycle leads to
a modification of the measured oscillation parameters. We present a specific
$\mu-\tau$ symmetric model in which the measured value of $\theta_{13}$ is
entirely generated by the DM interaction, and which reproduces the other
measured oscillation parameters. For a scalar DM particle lighter than
$10^{-15}$ eV, adiabatic solar neutrino propagation is maintained. A
suppression of the sensitivity to CP violation at long-baseline neutrino
experiments is predicted in this model. We find that DUNE cannot exclude the DM
scenario at more than $3\sigma$ C.L. for bimaximal, tribimaximal and hexagonal
mixing, while JUNO can rule it out at more than $6\sigma$ C.L. by precisely
measuring both $\theta_{12}$ and $\theta_{13}$.
