Liquid metals are promising heat transfer agents in new-generation nuclear power plants, such as fast-neutron reactors and hybrid tokamaks—fusion neutron sources (FNSs). We have been investigating hydrodynamics and heat exchange of liquid metals for many years, trying to reproduce the conditions close to those in fast reactors and fusion neutron sources. In the latter case, the liquid metal flow takes place in a strong magnetic field and strong thermal loads resulting in development of thermogravitational convection in the flow. In this case, quite dangerous regimes of magnetohydrodynamic (MHD) heat exchange not known earlier may occur that, in combination with other long-known regimes, for example, the growth of hydraulic drag in a strong magnetic field, make the possibility of creating a reliable FNS cooling system with a liquid metal heat carrier problematic. There exists a reasonable alternative to liquid metals in FNS, molten salts, namely, the melt of lithium and beryllium fluorides (Flibe) and the melt of fluorides of alkali metals (Flinak). Molten salts, however, are poorly studied media, and their application requires detailed scientific substantiation. We analyze the modern state of the art of studies in this field. Our contribution is to answer the following question: whether above-mentioned extremely dangerous regimes of MHD heat exchange detected in liquid metals can exist in molten salts. Experiments and numerical simulation were performed in order to answer this question. The experimental test facility represents a water circuit, since water (or water with additions for increasing its electrical conduction) is a convenient medium for laboratory simulation of salt heat exchange in FNS conditions. Local heat transfer coefficients along the heated tube, three-dimensional (along the length and in the cross section, including the viscous sublayer) fields of averaged temperature and temperature pulsations are studied. The probe method for measurements in a flow is described in detail. Experimental data are designated for verification of codes simulating heat exchange of molten salts.