We present new, high sensitivity VLA observations of HI in four dwarf galaxies (UGCA 292, GR8, DDO 210, and DDO 216) and we use these data to study interactions between star formation and the interstellar medium. HI velocity dispersions and line shapes in UGCA 292, GR8, and DDO 210 show that these three galaxies contain both warm and cool or cold HI phases. The presence of the cold neutral medium is indicated by a low-dispersion (3--6 km/s) HI component or by the Gauss-Hermite shape parameter h_4 > 0. Contrary to expectations, we find no trend between the incidence of the low-dispersion (colder) phase and the star formation rate in five dwarf galaxies. The colder HI phase may be a necessary ingredient for star formation, but it is clearly not sufficient. However, there is a global trend between the star formation rate of a galaxy and the incidence of asymmetric HI profiles. This trend probably reflects kinetic energy input from young massive stars. Numerical simulations show that the effects of rotational broadening (finite angular resolution) are minimal for these galaxies. Simulations are also used to estimate the errors in the column densities of the high-dispersion and the low-dispersion HI phases.