In this paper we present deep Arecibo H I and WIYN optical observations of Virgo Cluster dwarf elliptical galaxies. Based on this data we argue that a significant fraction of low-mass galaxies in the Virgo Cluster recently underwent evolution. Our new observations consist of H I 21 cm line observations for 22 classified dE galaxies with optical radial velocities consistent with membership in the Virgo Cluster. Cluster members VCC 390 and VCC 1713 are detected with H I masses M = 6 × 10⁷ and 8 × 10⁷ M, respectively, while M_{H i} values in the remaining 20 dE galaxies have upper limits as low as ~5 × 10⁵ M. We combine our results with those for 26 other Virgo Cluster dE galaxies with H I observations in the literature, seven of which have H I detection claims. New optical images from the WIYN telescope of five of these H I-detected dE galaxies, along with archival data, suggest that seven of the claimed detections are true H I detections, yielding a ≈15% detection rate. These H I-detected, classified dE galaxies are preferentially located near the periphery of the Virgo Cluster. Three Virgo dE galaxies have observed H I velocity widths greater than 200 km s^-1, possibly indicating the presence of a large dark matter content or transient extended H I. We discuss the possible origins of these objects and argue that they originate from field galaxies accreted onto high angular momentum orbits by Virgo in the last few Gyr. As a result of this, we argue, these galaxies are slowly transformed within the cluster by gradual gas-stripping processes, associated truncation of star formation, and passive fading of stellar populations. Low-mass, early-type cluster galaxies are therefore currently being produced as the product of cluster environmental effects. We utilize our results in a simple model to estimate the recent (past 1-3 Gyr) average mass accretion rate into the Virgo Cluster, deriving a value of ~ 50 M yr^-1.