We investigate the far-infrared properties of galaxies selected via deep, narrow-band imaging of the H$\alpha$ emission line in four redshift slices from $z=0.40$--$2.23$ over $\sim 1$deg$^2$ as part of the High-redshift Emission Line Survey (HiZELS). We use a stacking approach in the Herschel PACS/SPIRE bands, along with $850\,\mu$m imaging from SCUBA-2 to study the evolution of the dust properties of H$\alpha$-emitters selected above an evolving characteristic luminosity threshold, $0.2L^\star_{{\rm H}\alpha}(z)$. We investigate the relationship between the dust temperatures and the far-infrared luminosities of our stacked samples, finding that H$\alpha$-selection identifies cold, low-$L_{\rm IR}$ galaxies ($T_{\rm dust}\sim 14$k; $\log[L_{\rm IR}/{\rm L}_\odot]\sim 9.9$) at $z=0.40$, and more luminous, warmer systems ($T_{\rm dust}\sim 34$k; $\log[L_{\rm IR}/{\rm L}_\odot]\sim 11.5$) at $z=2.23$. Using a modified greybody model, we estimate "characteristic sizes" for the dust-emitting regions of HiZELS galaxies of $\sim 0.5$kpc, nearly an order of magnitude smaller than their stellar continuum sizes, which may provide indirect evidence of clumpy ISM structure. Lastly, we measure the dust masses from our far-IR SEDs along with metallicity-dependent gas-to-dust ratios ($\delta_{\rm GDR}$) to measure typical molecular gas masses of $\sim 10^{10}$M$_\odot$ for these bright H$\alpha$-emitters. The gas depletion timescales are shorter than the Hubble time at each redshift, suggesting probable replenishment of their gas reservoirs from the intergalactic medium. Based on the number density of H$\alpha$-selected galaxies, we find that typical star-forming galaxies brighter than $0.2L^{\star}_{{\rm H}\alpha}(z)$ host a significant fraction ($35\pm10$%) of the total gas content of the Universe, consistent with the predictions of the latest cosmological simulations.