We investigate the infrared (IR) emission of high-redshift ($zsim 6$), highly star-forming (${ {rm SFR} > 100}$ $M_{rm odot} {rm yr}^{-1}$) 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 (${M_{rm d}gtrsim 3times 10^7 M_{rm odot}}$) galaxies, $gtrsim 50-90 %$ of the UV radiation is obscured by dust inhomogeneities on scales ${gtrsim 100}$ pc. In runs with AGN, a clumpy, warm ($approx 250$ K) dust component co-exists with a colder ($approx 60$ K) and more diffuse one, heated by stars. Warm dust provides up to ${50 %}$ of the total IR luminosity, but only $lesssim 0.1 %$ 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 FIR. Our simulations successfully reproduce the observed SED of bright (${M_{rm UV}sim -26}$) ${zsim 6}$ quasars, and show that these objects are part of complex, dust-rich merging systems, containing multiple sources (accreting BHs 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 MIR properties of dusty star forming galaxies and to obtain good quality spectra of bright quasars at $zsim 6$. Finally, the MIR-to-FIR flux ratio of faint (${M_{rm UV}sim -24}$) AGN is ${>10times}$ 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.
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