The structure and dynamics of the hydrogen-bond network in heavy water (D2O) is studied as a function of the temperature using quantum dynamical simulations. Our approach combines an ab initio-based representation of the water interactions with an explicit quantum treatment of the molecular motion. A direct connection between the calculated linear and nonlinear vibrational spectra and the underlying molecular dynamics is made, which provides new insights into the rearrangement of the hydrogen-bond network in heavy water. A comparison with previous calculations on liquid H2O suggests that tunneling does not effectively contribute to the dynamics of the water hydrogen-bond network above the melting point. However, the effects of nuclear quantization are not negligible at all temperatures and become increasingly important near the melting point, which is in agreement with recent experimental analysis of the structural properties of liquid water as well as of the proton momentum distribution in supercooled water.