Context. Showing 1.4GHz flux densities in the range of a few to a few tens of mJy, infrared-faint radio sources (IFRS) are a type of galaxy characterised by faint or absent near-infrared counterparts and consequently extreme radio-to-infrared flux density ratios up to several thousand. Recent studies showed that IFRS are radio-loud active galactic nuclei (AGNs) at redshifts ≳2, potentially linked to high-redshift radio galaxies (HzRGs). Aims. This work explores the far-infrared emission of IFRS, providing crucial information on the star forming and AGN activity of IFRS. Furthermore, the data enable examining the putative relationship between IFRS and HzRGs and testing whether IFRS are more distant or fainter siblings of these massive galaxies. Methods. A sample of six IFRS was observed with the Herschel Space Observatory between 100 μm and 500 μm. Using these results, we constrained the nature of IFRS by modelling their broad-band spectral energy distribution (SED). Furthermore, we set an upper limit on their infrared SED and decomposed their emission into contributions from an AGN and from star forming activity. Results. All six observed IFRS were undetected in all five Herschel far-infrared channels (stacking limits: σ = 0.74 mJy at 100 μm, σ = 3.45 mJy at 500 μm). Based on our SED modelling, we ruled out the following objects to explain the photometric characteristics of IFRS: (a) known radio-loud quasars and compact steep-spectrum sources at any redshift; (b) starburst galaxies with and without an AGN and Seyfert galaxies at any redshift, even if the templates were modified; and (c) known HzRGs at z ≲ 10.5. We find that the IFRS analysed in this work can only be explained by objects that fulfil the selection criteria of HzRGs. More precisely, IFRS could be (a) known HzRGs at very high redshifts (z ≳ 10.5); (b) low-luminosity siblings of HzRGs with additional dust obscuration at lower redshifts; (c) scaled or unscaled versions of Cygnus A at any redshift; and (d) scaled and dust-obscured radio-loud quasars or compact steep spectrum sources. We estimated upper limits on the infrared luminosity, the black hole accretion rate, and the star formation rate of IFRS, which all agreed with corresponding numbers of HzRGs.