The X-ray diffraction, Raman and infrared spectroscopies and magnetic measurements were used to explore the correlated changes of the structure, lattice dynamics and magnetic properties of the LuFeO3 nanoparticles, which appear in dependence on their sintering temperature. We revealed a gradual substitution of the hexagonal phase by the orthorhombic phase in the nanoparticles, which sintering temperature increases from 700 C to 1100 C. The origin and stability of the hexagonal phase in the LuFeO3 nanoparticles is of the special interest, because the nanoparticle in the phase can be a room-temperature multiferroic with a weak ferromagnetic and pronounced structural and ferroelectric long-range ordering, while the antiferromagnetic and nonpolar orthorhombic phase is more stable in the bulk LuFeO3. To define the ranges of the hexagonal phase stability, we determine the bulk and interface energy densities of different phases from the comparison of the Gibbs model with experimental results. Using the Gibbs model parameters, we predict the influence of size effects and temperature on the structural and polar properties of the LuFeO3 nanoparticles. Analysis of the obtained results shows that the combination of the X-ray diffraction, Raman and infrared spectroscopy, magnetic measurements and theoretical modelling of structural and polar properties allows to establish the interplay between the phase composition, lattice dynamics and multiferroic properties of the LuFeO3 nanoparticles prepared in different conditions.