This is the fourth in our series of papers investigating radio-excess galaxies, which have radio emission associated with an active nucleus but which do not fit into the traditional categories of either radio-loud or radio-quiet active galaxies. In this paper, we present optical spectra of our sample of FIR-luminous radio-excess galaxies. Optical emission line diagnostics are used to determine the dominant source of the ionizing radiation. We find that radio excess is an excellent indicator of the presence of an active nucleus: the radio-excess sample contains a much higher fraction of AGN than samples selected on FIR luminosity alone, or using other criteria such as warm FIR colors. Several objects have ambiguous classifications and are likely to be composite objects with mixed excitation. The type of optical spectrum appears to be associated with the radio-loudness: radio-loud objects may be more `pure' AGN than radio-intermediate objects. We find strong evidence for interaction between the radio plasma and the surrounding gas. The jet energy fluxes of the radio-excess objects, inferred from the [O III] luminosities, are lower than in powerful radio sources, consistent with our previous results. We conclude that the jets of radio-intermediate sources are intrinsically weaker than those in sources with more powerful radio emission. A significant fraction of the sample spectra show post-starburst stellar continuum, with A-star absorption lines, consistent with the large fraction of merging or disturbed host galaxies in the sample. The ages of the radio sources are significantly less than those of A stars indicating that, if the radio sources are associated with merging activity, there is a delay between the interaction and the initiation of the radio activity. (Abridged.)

Accepted for publication in AJ; version with high resolution figures available from http://www.cis.rit.edu/~clbsps/papers/paper4.pdf