  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.)
