Doping of semiconductor nanocrystals is an emerging tool to control their properties and has recently received increased interest as the means to characterize the impurities and their effect on the electronic characteristics of the nanocrystal evolve. We present a temperature-dependent Raman scattering study of Cu-doped InAs nanocrystals observing changes in the relative scattering intensities of the different modes upon increased dopant concentrations. First, the longitudinal optical (LO) phonon overtone mode is suppressed, indicating weakening of the coupling strength related to the effect of screening by the free electrons. Second, the transverse optical (TO) mode is relatively enhanced compared to the LO mode, which is attributed to the appearance of a coupled phonon-plasmon mode analogous to observations for n-type doped bulk InAs. These signatures indicate that the Cu impurities serve as active dopants and occupy an impurity-related pseudo sub-band akin to the heavy doping limit.