We investigate the far-infrared ( far-IR ) 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 far-IR bands , along with 850 \mu m imaging from SCUBA-2 and Very Large Array ( VLA ) 1.4 GHz imaging to study the evolution of the dust properties of H \alpha -emitters selected above an evolving characteristic luminosity threshold , 0.2 L ^ { \star } _ { { H } \alpha } ( z ) . We investigate the relationship between the dust temperatures , T _ { dust } , and the far-infrared luminosities , L _ { IR } of our stacked samples , finding that our H \alpha -selection identifies cold , low- L _ { IR } galaxies ( T _ { dust } \sim 14 k ; \log [ L _ { IR } / { L } _ { \odot } ] \sim 9.9 ) at z = 0.40 , and more luminous , warmer systems ( T _ { dust } \sim 34 k ; \log [ L _ { IR } / { L } _ { \odot } ] \sim 11.5 ) at z = 2.23 . Using a modified grey-body model , we estimate “ characteristic sizes ” for the dust-emitting regions of H \alpha -selected 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 use measurements of the dust masses from our far-IR stacking along with metallicity-dependent gas-to-dust ratios ( \delta _ { GDR } ) to measure typical molecular gas masses of \sim 1 \times 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 0.2 L ^ { \star } _ { { H } \alpha } ( z ) comprise a significant fraction ( 35 \pm 10 % ) of the total gas content of the Universe , consistent with the predictions of the latest state-of-the-art cosmological simulations .