Couplings between light scalar (fuzzy) 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 $\theta_{13}$.