We analyze the problem of eliminating finite-size errors from quantum Monte Carlo (QMC) energy data. We demonstrate that both (i) adding a recently proposed [ S. Chiesa et al. Phys. Rev. Lett. 97 076404 (2006)] finite-size correction to the Ewald energy and (ii) using the model periodic Coulomb (MPC) interaction [ L. M. Fraser et al. Phys. Rev. B 53 1814 (1996); P. R. C. Kent et al. Phys. Rev. B 59 1917 (1999); A. J. Williamson et al. Phys. Rev. B 55 R4851 (1997)] are good solutions to the problem of removing finite-size effects from the interaction energy in cubic systems provided the exchange-correlation (XC) hole has converged with respect to system size. However, we find that the MPC interaction distorts the XC hole in finite systems, implying that the Ewald interaction should be used to generate the configuration distribution. The finite-size correction of Chiesa et al. Phys. Rev. Lett. 97 076404 (2006) is shown to be incomplete in systems of low symmetry. Beyond-leading-order corrections to the kinetic energy are found to be necessary at intermediate and high densities; we investigate the effect of adding such corrections to QMC data for the homogeneous electron gas. We analyze finite-size errors in two-dimensional systems and show that the leading-order behavior differs from that which has hitherto been supposed. We compare the efficiencies of different twist-averaging methods for reducing single-particle finite-size errors and we examine the performance of various finite-size extrapolation formulas. Finally, we investigate the system-size scaling of biases in diffusion QMC.