We use archival Herschel data to examine the singly ionized carbon ([C II]) content of 14 star-forming dwarf galaxies in the Virgo cluster. We use spectral energy distribution fits to far-infrared, mid-infrared, near-infrared, optical, and ultraviolet data to derive the total infrared continuum (TIR) for these galaxies. We compare the [C II]/TIR ratio for dwarf galaxies in the central part of Virgo to those in the southern part of the cluster and to galaxies with similar TIR luminosities and metallicities in the Herschel Dwarf Galaxy Survey (DGS) sample of field dwarf galaxies to look for signs of [C II] formation independent of star formation. Our analysis indicates that the sample of Virgo dwarfs in the central part of the cluster has significantly higher values of [C II]/TIR than the sample from the southern part of the cluster and the sample from the DGS, while the southern sample is consistent with the DGS. This [C II]/TIR excess implies that a significant fraction of the [C II] in the dwarf galaxies in the cluster center has an origin other than star formation and is likely to be due to environmental processes in the central part of the cluster. We also find a surprisingly strong correlation between [C II]/TIR and the local ram pressure felt by the dwarf galaxies in the cluster. In this respect, we claim that the excess [C II] we see in these galaxies is likely to be due to formation in ram-pressure shocks.