The current study investigates the effect of heating defined via Richardson number on the stability of swirling flows via linear stability analysis and direct numerical simulations. Such flows are common in combustion and mixing applications and are simple models for atmospheric flows such as fire whirls and dust devils. The linear stability characteristics of azimuthal wavenumbers m = 1-5 are investigated at a fixed Reynolds number of 500 and at three inlet swirl angles where the non-buoyant (without heating) flow shows linear stability characteristics different from each other. The results show that the heating may initially have a stabilising effect but with more heating, the flow ultimately becomes unstable to perturbations. The growth rate of the leading eigenmode agrees with the predictions of three-dimensional direct numerical simulations. The centre-line axial velocity is increased noticeably with heating, indicating much larger axial momentum in unstable buoyant flows than non-buoyant flows.