Molecular hydrogen is the most abundant molecular species in the universe. While no doubts exist that it is mainly formed on the interstellar dust grain surfaces, many details of this process remain poorly known. In this work, we focus on the fate of the energy released by the H₂ formation on the dust icy mantles: how it is partitioned between the substrate and the newly formed H₂, a process that has a profound impact on the interstellar medium. We carried out state-of-the-art ab initio molecular dynamics simulations of H₂ formation on periodic crystalline and amorphous ice surface models. Our calculations show that up to two-thirds of the energy liberated in the reaction (~300 kJ mol^-1 ~3.1 eV) is absorbed by the ice in less than 1 ps. The remaining energy (~140 kJ mol^-1 ~1.5 eV) is kept by the newly born H₂. Since it is 10 times larger than the H₂ binding energy on the ice, the new H₂ molecule will eventually be released into the gas phase. The ice water molecules within ~4 Å from the reaction site acquire enough energy, between 3 and 14 kJ mol^-1 (360-1560 K), to potentially liberate other frozen H₂ and, perhaps, frozen CO molecules. If confirmed, the latter process would solve the long standing conundrum of the presence of gaseous CO in molecular clouds. Finally, the vibrational state of the newly formed H₂ drops from highly excited states (ν = 6) to low (ν ≤ 2) vibrational levels in a timescale of the order of picoseconds.