We quantify the luminosity contribution of active galactic nuclei (AGNs) to the 12 mu m, mid-infrared (MIR; 5-38 mu m), and total IR (5-1000 mu m) emission in the local AGNs detected in the all-sky 70 month Swift/Burst Alert Telescope (BAT) ultrahard X-ray survey. We decompose the IR spectral energy distributions (SEDs) of 587 objects into the AGN and starburst components using templates for an AGN torus and a star-forming galaxy. This enables us to recover the emission from the AGN torus including the low-luminosity end, down to log(L14-150/erg s(-1)) similar or equal to 41, which typically has significant host galaxy contamination. The sample demonstrates that the luminosity contribution of the AGN to the 12 mu m, the MIR, and the total IR bands is an increasing function of the 14-150 keV luminosity. We also find that for the most extreme cases, the IR pure-AGN emission from the torus can extend up to 90 mu m. The total IR AGN luminosity obtained through the IR SED decomposition enables us to estimate the fraction of the sky obscured by dust, i.e., the dust covering factor. We demonstrate that the median dust covering factor is always smaller than the median X-ray obscuration fraction above an AGN bolometric luminosity of log(L-bol((AGN))/erg s(-1)) similar or equal to 42.5. Considering that the X-ray obscuration fraction is equivalent to the covering factor coming from both the dust and gas, this indicates that an additional neutral gas component, along with the dusty torus, is responsible for the absorption of X-ray emission.