High-redshift submillimetre-bright galaxies identified by blank field surveys at millimetre and submillimetre wavelengths appear in the region of the Infra Red Array Camera (IRAC) colour–colour diagrams previously identified as the domain of luminous active galactic nuclei (AGNs). Our analysis using a set of empirical and theoretical dusty starburst spectral energy distribution (SED) models shows that power-law continuum sources associated with hot dust heated by young (≲100 Myr old), extreme starbursts at z > 2 also occupy the same general area as AGNs in the IRAC colour–colour plots. A detailed comparison of the IRAC colours and SEDs demonstrates that the two populations are distinct from each other, with submillimetre-bright galaxies having a systematically flatter IRAC spectrum (≳1 mag bluer in the observed [4.5]–[8.0] colour). Only about 20 per cent of the objects overlap in the colour–colour plots, and this low fraction suggests that submillimetre galaxies powered by a dust-obscured AGN are not common. The red infrared colours of the submillimetre galaxies are distinct from those of the ubiquitous foreground IRAC sources, and we propose a set of infrared colour selection criteria for identifying SMG counterparts that can be used even in the absence of radio or Multiband Imaging Photometer for Spitzer (MIPS) 24 μm data.