Measurements of the response of LaBr3(Ce) to 2.5 MeV neutrons have been carried out at the Frascati Neutron Generator and at tokamak facilities with deuterium plasmas. The observed spectrum has been interpreted by means of a Monte Carlo model. It is found that the main contributor to the measured response is neutron inelastic scattering on (79)Br, (81)Br, and (139)La. An extrapolation of the count rate response to 14 MeV neutrons from deuterium-tritium plasmas is also presented. The results are of relevance for the design of γ-ray diagnostics of fusion burning plasmas.