ABSTRACT We investigate the infrared (IR) emission of high-redshift (z ∼ 6), highly star-forming (${{\rm SFR}\gt 100\,{\rm M}_{\odot }\, {\rm yr}^{-1}}$) galaxies, with/without active galactic nuclei (AGN), using a suite of cosmological simulations featuring dust radiative transfer. Synthetic spectral energy distributions (SEDs) are used to quantify the relative contribution of stars/AGN to dust heating. In dusty (Md ≳ 3 × 107 M⊙) galaxies, ≳50–90 per cent of the ultraviolet (UV) radiation is obscured by dust inhomogeneities on scales ≳100 pc. In runs with AGN, a clumpy, warm (≈250 K) dust component coexists with a colder (≈60 K) and more diffuse one, heated by stars. Warm dust provides up to ${50 {{\ \rm per\ cent}}}$ of the total infrared (IR) luminosity, but only ${\lesssim}0.1 {{\ \rm per\ cent}}$ of the total mass content. The AGN boosts the MIR flux by 10–100 times with respect to star-forming galaxies, without significantly affecting the far-IR. Our simulations successfully reproduce the observed SED of bright (MUV ∼ −26) z ∼ 6 quasars, and show that these objects are part of complex, dust-rich merging systems, containing multiple sources (accreting black holes and/or star-forming galaxies) in agreement with recent HST and ALMA observations. Our results show that the proposed ORIGINS missions will be able to investigate the mid-IR (MIR) properties of dusty star-forming galaxies and to obtain good-quality spectra of bright quasars at z ∼ 6. Finally, the MIR-to-FIR flux ratio of faint (MUV ∼ −24) AGN is >10 times higher than for normal star-forming galaxies. This implies that combined JWST/ORIGINS/ALMA observations will be crucial to identify faint and/or dust-obscured AGN in the distant Universe.