We present observations of six late-type members of the young cluster IC 348 detected at 24 μm with the Multiband Imaging Photometer for Spitzer (MIPS). At least four of the objects are probably substellar. Combining these data with ground-based optical and near-infrared photometry and complementary observations with the Infrared Array Camera (IRAC), we have modeled the spectral energy distributions using detailed models of irradiated accretion disks. We are able to fit the observations with models using a range of maximum grain sizes from ISM-type dust to grains as large as 1 mm. Two objects show a lack of excess emission at wavelengths shortward of 5.8-8 μm but significant excess at longer wavelengths, indicative of large optically thin or evacuated inner holes. Our models indicate an inner hole of radius ~0.5-0.9 AU for the brown dwarf L316; this is the first brown dwarf with evidence for an AU-scale inner disk hole. We examine several possible mechanisms for the inner disk clearing in this case, including photoevaporation and planet formation.