The lines on the characteristic absorption spectra of a particular atom represent the energy differences between the potential characteristic energy levels that its valence electron(s) can occupy and are produced by the electrons actually absorbing the photons with those specific wavelengths and jumping up to the higher energy levels. Positive ions, by definition, have lost valence electrons and therefore the electrons aren’t there to absorb the photons.
The title is a bit of a blanket statement though and not completely true - there are usually several characteristic lines in any atom’s spectra because multiple electrons are each making jumps between different characteristic energy levels, so even if the main valence electron(s) are absent in an ion, there may be other electrons left that are able to absorb photons of other wavelengths but you wouldn’t get the atom’s full spectra.