  We investigate the infrared (IR) emission of high-redshift ($z\sim 6$),
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 (${M_{\rm d}\gtrsim 3\times 10^7 M_{\rm
\odot}}$) 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 (${M_{\rm UV}\sim -26}$) ${z\sim 6}$
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 $z\sim 6$. Finally, the MIR-to-FIR flux ratio of faint
(${M_{\rm UV}\sim -24}$) 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.
