Despite the common picture of an early-type dwarf (dE) as a quiescent galaxy with no star formation and little gas, we identify 23 dEs that have blue central colors caused by recent or ongoing star formation in our sample of 476 Virgo cluster dEs. In addition, 14 objects that were mostly classified as (candidate) BCDs have similar properties. Among the certain cluster members, the dEs with blue centers reach a fraction of more than 15% of the dE population at brighter (m B ≤ 16) magnitudes. A spectral analysis of the centers of 16 galaxies reveals in all cases an underlying old population that dominates the mass, with M old ≥ 90% for all but one object. Therefore the majority of these galaxies will appear like ordinary dEs within ∼ one Gigayear or less after the last episode of star formation. Their overall gas content is less than that of dwarf irregular galaxies, but higher than that of ordinary dEs. Their flattening distribution suggests the shape of a thick disk, similar to what has been found for dEs with disk features in Paper I of this series. Their projected spatial distribution shows no central clustering, and their distribution with projected local density follows that of irregular galaxies, indicative of an unrelaxed population. This is corroborated by their velocity distribution, which displays two side peaks characteristic of recent infall. We discuss possible formation mechanisms (ram-pressure stripping, tidally induced star formation, harassment) that might be able to explain both the disk shape and the central star formation of the dEs with blue centers.