  We investigate the constraints on total neutrino mass in the scenario of
vacuum energy interacting with cold dark matter. We focus on two typical
interaction forms, i.e., $Q=\beta H\rho_{\rm c}$ and $Q=\beta H\rho_{\Lambda}$.
To avoid the occurrence of large-scale instability in interacting dark energy
cosmology, we adopt the parameterized post-Friedmann approach to calculate the
perturbation evolution of dark energy. We employ observational data, including
the Planck cosmic microwave background temperature and polarization data,
baryon acoustic oscillation data, a JLA sample of type Ia supernovae
observation, direct measurement of the Hubble constant, and redshift space
distortion data. We find that, compared with those in the $\Lambda$CDM model,
much looser constraints on $\sum m_{\nu}$ are obtained in the $Q=\beta
H\rho_{\rm c}$ model, whereas slightly tighter constraints are obtained in the
$Q=\beta H\rho_{\Lambda}$ model. Consideration of the possible mass hierarchies
of neutrinos reveals that the smallest upper limit of $\sum m_{\nu}$ appears in
the degenerate hierarchy case. By comparing the values of $\chi^2_{\rm min}$,
we find that the normal hierarchy case is favored over the inverted one. In
particular, we find that the difference $\Delta \chi^2_{\rm min} \equiv
\chi^2_{\rm IH; min}-\chi^2_{\rm NH; min}> 2$ in the $Q=\beta H\rho_{\rm c}$
model. In addition, we find that $\beta=0$ is consistent with the current
observations in the $Q=\beta H\rho_{\rm c}$ model, and $\beta < 0$ is favored
at more than the $1\sigma$ level in the $Q=\beta H\rho_{\Lambda}$ model.
