We constrain the neutrino mass in the scenario of vacuum energy interacting with cold dark matter by using current cosmological observations. To avoid the large-scale instability problem in interacting dark energy models, we employ the parameterized post-Friedmann (PPF) approach to do the calculation of perturbation evolution, for the $Q=β Hρ_{\rm c}$ and $Q=β Hρ_{Λ}$ models. According to the constraint results, we find that $β>0$ at more than $1σ$ level for the $Q=β Hρ_{\rm c}$ model, which indicates that cold dark matter decays into vacuum energy; while $β=0$ is consistent with the current data at $1σ$ level for the $Q=β Hρ_{Λ}$ model. Taking the $Λ$CDM model as a baseline model, we find that a smaller upper limit, $∑ m_{ν}