The temporal stability of a two-layer plane Couette flow with an interfacial heat source/sink is studied, in the presence of a viscosity stratification, predicting unstable streamwise and spanwise perturbations. Based on the wavenumber at onset, the perturbations are further classified as long-wave and finite-wave, where the long-wave mode is investigated using asymptotic and numerical analyses. The finite-wave instability is expected to be relevant for highly viscous fluids. Generally, a heat source (sink) at the interface is found to have a stabilising (destabilising) effect on the flow. However, if the rate of heat release exceeds a certain threshold, a new instability emerges, referred to here as an 'explosive' instability, characterised by an unbounded growth rate of the perturbations. The interaction between the viscosity stratification, thermocapillarity and inertia leads to the emergence of another new mode of streamwise instability, termed the 'interaction' instability. The spanwise perturbations possess higher growth rates than the streamwise perturbations for the heat sink along the interface when inertia has a stabilising influence, and vice versa. The heat sink leads to the emergence of thermocapillary instability by generating a negative temperature gradient across the fluids. In contrast, a heat source increases the temperature of the interface, resulting in a positive temperature gradient across the fluids, leading to the suppression of the instabilities by the thermocapillary effect. If the heat source provides energy at a faster rate than the energy lost due to Marangoni convection and heat diffusion, then it leads to an accumulation of energy, which presumably leads to the explosive instability. © 2022 The Author(s). Published by Cambridge University Press.