General aspects of the finite-size scaling behavior of constrained free energy barriers between coexisting phases are discussed and illustrated with the two-dimensional Ising model. Particular attention is given to a method introduced by Binder [Phys. Rev. A 25, 1699 (1982)], in the context of lattice models, for extracting the interfacial tension based on an extrapolation to the thermodynamic limit of the barrier height divided by a quantity related to the total interfacial area. These ideas are then applied to a 3D Lennard-Jones system. The height of the constrained free energy barrier Delta F(V) separating coexisting gas and liquid phases in a Lennard-Jones fluid is determined for various values of the temperature T and number of particles N, using an isothermal-isobaric Monte Carlo simulation in conjunction with biased sampling and reweighting techniques. The critical temperature T-c=1.32 is readily established even from the results for very small systems by observing the value of T for which Delta F(V) is essentially independent of N. The extrapolation of the effective interfacial tension to the thermodynamic limit using Binder’s method is complicated because the data display a non-monotonic N-dependence, similar to that observed recently in the 3D Ising model [Berg, Hansmann, and Neuhaus, Z. Phys. B 90, 229 (1993)]. This behavior appears to arise at least in part from interactions between the two interfaces in the periodic simulation box. A self-consistent fit of all of the effective interfacial tension data is made based on a finite-size scaling ansatz and assuming the standard critical exponent mu=1.26; this leads to a value of gamma(0)=2.79 in the universal expression for the surface tension gamma=gamma(0)(1-T/T-c)(mu), which is in reasonable agreement with the expected value. Definitive extrapolations will require substantially larger simulations.