We present high spatial and spectral resolution HI synthesis observations of seven compact dwarf irregular galaxies to investigate the relationship between gas distribution, kinematics, and star formation activity in low mass galaxies. The compact dwarf irregular class contains a heterogeneous mixture of galaxies. Three of the galaxies in this sample are likely counterparts of starbursting dwarf galaxies at later (or earlier) evolutionary stages; these three galaxies have similar morphological and kinematic properties as the starbursting dwarf galaxy class. The remaining four galaxies in this sample have unusually large HI envelopes that extend 2 - 7 times their optical diameters. For two of these galaxies, the extended gas distributions are reminiscent of tidal tails, which suggests that the star formation activity in these galaxies may have been triggered by previous interactions (despite their current isolation). The remaining two galaxies have warped gas disks, but much more regular morphology. These observations support the hypothesis that some compact starbursting galaxies will retain the majority of their interstellar medium after the starburst episode; thus, post-burst galaxies may be found as a subset of the general class of gas-rich dwarf irregular galaxies.