The concept of a tokamak-based stationary fusion neutron source (FNS) for scientific research (neutron diffraction, etc.), tests of structural materials for future fusion reactors, nuclear waste transmutation, fission reactor fuel production, and control of subcritical nuclear systems (fusion–fission hybrid reactor) is being developed in Russia. The fuel cycle system is one of the most important systems of FNS that provides circulation and reprocessing of the deuterium–tritium fuel mixture in all fusion reactor systems: the vacuum chamber, neutral injection system, cryogenic pumps, tritium purification system, separation system, storage system, and tritium-breeding blanket. The existing technologies need to be significantly upgraded since the engineering solutions adopted in the ITER project can be only partially used in the FNS (considering the capacity factor higher than 0.3, tritium flow up to 200 m3Pa/s, and temperature of reactor elements up to 650°C). The deuterium–tritium fuel cycle of the stationary FNS is considered. The TC-FNS computer code developed for estimating the tritium distribution in the systems of FNS is described. The code calculates tritium flows and inventory in tokamak systems (vacuum chamber, cryogenic pumps, neutral injection system, fuel mixture purification system, isotope separation system, tritium storage system) and takes into account tritium loss in the fuel cycle due to thermonuclear burnup and β decay. For the two facility versions considered, FNS-ST and DEMO-FNS, the amount of fuel mixture needed for uninterrupted operation of all fuel cycle systems is 0.9 and 1.4 kg, consequently, and the tritium consumption is 0.3 and 1.8 kg per year, including 35 and 55 g/yr, respectively, due to tritium decay.