We present James Webb Space Telescope/Mid-Infrared Instrument imaging of eight nearby active galactic nuclei (AGN) from the GATOS (Galactic Activity, Torus, and Outflow Survey) survey to investigate the physical conditions of extended dust in their narrow-line regions (NLRs). In four galaxies (ESO 428─G14, NGC 4388, NGC 3081, and NGC 5728), we detect spatially resolved dust structures extending ${\sim}$100─200 pc along the NLR. In these systems, we find a strong link between the morphology of the dust, the radio ejecta, and the coronal [Si VI] emission, implying that dust carries imprints of the processes shaping the NLR. Using spatially resolved spectral energy distributions, we show that dust in the NLR has systematically steeper slopes than star-forming clumps. This dust emits at temperatures in the range $150 \text{-} 220\, \rm K$, at a distance of $\sim$150 pc from the nucleus. Using simple models, we show that, even under optimistic assumptions of grain size and AGN luminosity, the excess mid-infrared emission cannot be explained by AGN illumination alone. We interpret this excess heating as in situ. We show that shocks with velocities $v_{\rm shock} \sim 200 \text{-} 400 \, \rm km \, s^{-1}$ in dense gas can close this gap, and in some cases even account for the total observed emission. This, combined with multiple lines of evidence for shocks in these regions, supports a scenario in which shocks not only coexist with dust but may be playing a key role in heating it. Our findings reveal shocks may be an important and previously overlooked driver of extended dust emission in the central hundreds of parsecs in AGN.