We present 5-40 μm spectroscopy of 41 planetary nebulae (PNe) in the Magellanic Clouds, observed with the Infrared Spectrograph on board the Spitzer Space Telescope. The spectra show the presence of a combination of nebular emission lines and solid state features from dust, superimposed on the thermal IR continuum. By analyzing the 25 LMC and 16 SMC PNe in our sample we found that the IR spectra of 14 LMC and four SMC PNe are dominated by nebular emission lines, while the other spectra show solid state features. We observed that the solid state features are compatible with carbon-rich dust grains (SiC, polycyclic aromatic hydrocarbons [PAHs], etc.) in all cases but three PNe, which show oxygen-rich dust features. The frequency of carbonaceous dust features is generally higher in LMC than in SMC PNe. The spectral analysis allowed the correlations of the dust characteristics with the gas composition and morphology, and the properties of the central stars. We found that (1) all PNe with carbonaceous dust features have C/O>1, none of these being bipolar or otherwise highly asymmetric; (2) all PNe with oxygen-rich dust features have C/O<1, with probable high-mass progenitors if derived from single-star evolution (these PNe are either bipolar or highly asymmetric); (3) the dust temperature tracks the nebular and stellar evolution; and (4) the dust production efficiency depends on metallicity, with low-metallicity environments not favoring dust production. Based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.