Two types of dust disks have been discovered around white dwarfs (WDs): small dust disks within the Roche limits of their WDs and large dust disks around hot WDs extending to radial distances of 10-10² AU. The majority of the latter WDs are central stars of planetary nebulae (CSPNs). We have therefore used archival Spitzer Infrared Array Camera (IRAC) and Multiband Imaging Photometer for Spitzer (MIPS) observations of PNs to search for CSPNs with IR excesses and to make a comparative investigation of dust disks around stars at different evolutionary stages. We have examined available images of 72 resolved PNs in the Spitzer archive and found 56 of them large enough for the CSPN to be resolved from the PN. Among these, only 42 CSPNs are visible in IRAC and/or MIPS images and selected for photometric measurements. From the spectral energy distributions (SEDs) of these CSPNs, we find 19 cases with clear IR excess. Of these, seven are [WC]-type stars, two have apparent visual companions that account for the observed excess emission, two are symbiotic CSPNs, and in eight cases the IR excess originates from an extended emitter, likely a dust disk. For some of these CSPNs, we have acquired follow-up Spitzer MIPS images, Infrared Spectrograph spectra, and Gemini NIRI and Michelle spectroscopic observations. The SEDs and spectra show a great diversity in the emission characteristics of the IR excesses, which may imply different mechanisms responsible for the excess emission. For CSPNs whose IR excesses originate from dust continuum, the most likely dust production mechanisms are (1) breakup of bodies in planetesimal belts through collisions and (2) formation of circumstellar dust disks through binary interactions. A better understanding of post-asymptotic giant branch binary evolution as well as debris disk evolution along with its parent star is needed to distinguish between these different origins. Future observations to better establish the physical parameters of the dust disks and the presence of companions are needed for models to discern between the possible dust production mechanisms.