We present Spitzer 4.5 and 8.0 μm imaging of 15 Local Group and nearby dwarf galaxies. We find that the diffuse 8 μm emission is spatially correlated with regions of active star formation. Our sample spans a range of >1 dex in nebular metallicity and 3 orders of magnitude in current star formation rate, allowing us to examine the dependence of emission from hot dust and PAHs on these parameters. We detect prominent diffuse 8 μm emission from the four most luminous galaxies in the sample (IC 1613, IC 5152, NGC 55, and NGC 3109) and only very low surface brightness emission from four others (DDO 216, Sextans A, Sextans B, and WLM). These are the first spatially resolved images of diffuse 8 μm emission from such low-metallicity objects [12 + log(O/H) ~ 7.5]. We observe correlations of this emission with the current star formation rate and the nebular metallicity of these galaxies. However, we also see evidence suggesting that other processes may also have a significant effect on the generation of this emission. These systems all have evidence for old and intermediate-age star formation; thus, the lack of diffuse 8 μm emission cannot be attributed to low galaxy ages. Also, winds cannot explain the paucity of this emission, since high-resolution imaging of the neutral gas in these objects shows no evidence of blowout. We propose that the lack of diffuse 8 μm emission in low-metallicity systems may be due to the destruction of dust grains by supernova shocks, assuming a long timescale to regrow dust. It is likely that the observed weak emission is at least partly due to a general absence of dust (including PAHs), in agreement with their low metallicities.