We present new mid- to far-infrared images of the two dwarf compact elliptical galaxies that are satellites of M31, NGC 185 and NGC 147, obtained with the Spitzer Space Telescope. Spitzers high sensitivity and spatial resolution enable us for the first time to look directly into the detailed spatial structure and properties of the dust in these systems. The images of NGC 185 at 8 and 24 micron display a mixed morphology characterized by a shell-like diffuse emission region surrounding a central concentration of more intense infrared emission. The lower resolution images at longer wavelengths show the same spatial distribution within the central 50 but beyond this radius, the 160 micron emission is more extended than that at 24 and 70micron. On the other hand, the dwarf galaxy NGC 147 located only a small distance away from NGC 185 shows no significant infrared emission beyond 24 micron and therefore its diffuse infrared emission is mainly stellar in origin. For NGC 185, the derived dust mass based on the best fit to the spectral energy distribution is 1.9e3 Msol, implying a gas mass of ~3.0e5 Msol. These values are in agreement with those previously estimated from infrared as well as from CO and HI observations and are consistent with the predicted mass return from dying stars based on the last burst of star formation ~1.0e9 yr ago. Based on the 70 to 160micron flux density ratio, we estimate a temperature for the dust of ~17K. For NGC 147, we obtain an upper limit for the dust mass of 4.5e2 Msol at 160 micron (assuming a temperature of ~20K), a value consistent with the previous upper limit derived using ISO observations of this galaxy. In the case of NGC 185, we also present full 5-38 micron low-resolution (R~100) spectra of the main emission regions.