In this paper, we first review general quantum mechanical limits on the sensitivity of heterodyne receivers. The main aim of the paper is to explore the quantum-noise (QN) properties of hot electron bolometric (HEB) mixers. HEB mixers have a characteristic feature not found in other mixers: based on the "hot-spot" model, the conversion loss varies along the length dimension of the bolometer, and some sections of the bolometer are essentially passive, in which little frequency conversion occurs. We analyze a quantitative distributed quantum-noise model of the HEB mixer, making use of simulated hot-spot model data, that takes into account the continuous variation of the sensitivity along the bolometer bridge. An expression for the HEB receiver noise temperature, including optical input loss, is derived. We find that the predicted double-sideband receiver noise temperature agrees well with the available measured data (up to 5.3 THz). The results of our analysis suggest that QN and classical HEB noise contribute about equally at 3 THz, while at higher terahertz frequencies QN dominates. QN thus appears to show measurable effects in existing HEB mixers and will be even more important to take into account as HEB mixers continue to be developed for higher terahertz frequencies.