The self-interaction error (SIE) is one of the major drawbacks of practical exchange-correlation functionals for Kohn-Sham density functional theory. Despite this, the use of methods that explicitly remove SIE from approximate density functionals is scarce in the literature due to their relatively high computational cost and lack of consistent improvement over standard modern functionals. In this article we assess the performance of a novel approach recently proposed by Pederson, Ruzsinszky, and Perdew [J. Chem. Phys. 2014, 140, 121103] for performing self-interaction free calculations in density functional theory based on Fermi orbitals. To this end, we employ test sets consisting of reaction energies that are considered particularly sensitive to SIE. We found that the parameter-free Fermi-Lowdin orbital self-interaction correction method combined with the standard local spin density approximation (LSDA) and Perdew-Burke-Ernzerhof (PBE) functionals gives a much better estimate of reaction energies compared to their parent LSDA and PBE functionals for most of the reactions in these two sets. They also perform on par with the global PBE0 and range-separated LC-omega PBE hybrids, which partially eliminate the SIE by including Hartree-Fock exchange. This shows the potential of the Fermi-Lowdin orbital self-interaction correction (FLOSIC) method for practical density functional calculations without SIE.