  The impact of dark matter-neutrino interactions on the measurement of the
cosmological parameters has been investigated in the past in the context of
massless neutrinos exclusively. Here we revisit the role of a neutrino-dark
matter coupling in light of ongoing cosmological tensions by implementing the
full Boltzmann hierarchy for three massive neutrinos. Our tightest 95% CL upper
limit on the strength of the interactions, parameterized via $u_\chi
=\frac{\sigma_0}{\sigma_{Th}}\left(\frac{m_\chi}{100 \text{GeV}}\right)^{-1}$,
is $u_\chi\leq3.34 \cdot 10^{-4}$, arising from a combination of Planck TTTEEE
data, Planck lensing data and SDSS BAO data. This upper bound is, as expected,
slightly higher than previous results for interacting massless neutrinos, due
to the correction factor associated with neutrino masses. We find that these
interactions significantly relax the lower bounds on the value of $\sigma_8$
that is inferred in the context of $\Lambda$CDM from the Planck data, leading
to agreement within 1-2$\sigma$ with weak lensing estimates of $\sigma_8$, as
those from KiDS-1000. However, the presence of these interactions barely
affects the value of the Hubble constant $H_0$.
