We investigate the impacts of dark energy on constraining massive (active/sterile) neutrinos in interacting dark energy (IDE) models by using the current observations. We employ two typical IDE models, the interacting $w$ cold dark matter (I$w$CDM) model and the interacting holographic dark energy (IHDE) model, to make an analysis. To avoid large-scale instability, we use the parameterized post-Friedmann approach to calculate the cosmological perturbations in the IDE models. The cosmological observational data used in this work include the Planck cosmic microwave background (CMB) anisotropies data, the baryon acoustic oscillation data, the type Ia supernovae data, the direct measurement of the Hubble constant, the weak lensing data, the redshift-space distortion data, and the CMB lensing data. We find that the dark energy properties could influence the constraint limits of active neutrino mass and sterile neutrino parameters in the IDE models. We also find that the dark energy properties could influence the constraints on the coupling strength parameter $beta$, and a positive coupling constant, $beta>0$, can be detected at the $2.5sigma$ statistical significance for the IHDE+$ u_s$ model by using the all-data combination. In addition, we also discuss the Hubble tension issue in these scenarios. We find that the $H_0$ tension can be effectively relieved by considering massive sterile neutrinos, and in particular in the IHDE+$ u_s$ model the $H_0$ tension can be reduced to be at the $1.28sigma$ level.
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