21-cm observations have the potential to revolutionize our understanding of the high-redshift Universe. Whilst extremely bright radio continuum foregrounds exist at these frequencies, their spectral smoothness can be exploited to allow efficient foreground subtraction. It is well known that – regardless of other instrumental effects – this removes power on scales comparable to the survey bandwidth. We investigate associated systematic biases. We show that removing line-of-sight fluctuations on large scales aliases into suppression of the 3D power spectrum across a broad range of scales. This bias can be dealt with by correctly marginalizing over small wavenumbers in the 1D power spectrum; however, the unbiased estimator will have unavoidably larger variance. We also show that Gaussian realizations of the power spectrum permit accurate and extremely rapid Monte Carlo simulations for error analysis; repeated realizations of the fully non-Gaussian field are unnecessary. We perform Monte Carlo maximum likelihood simulations of foreground removal which yield unbiased, minimum variance estimates of the power spectrum in agreement with Fisher matrix estimates. Foreground removal also distorts the 21-cm probability distribution function (PDF), reducing the contrast between neutral and ionized regions, with potentially serious consequences for efforts to extract information from the PDF. We show that it is the subtraction of large-scale modes which is responsible for this distortion, and that it is less severe in the earlier stages of reionization. It can be reduced by using larger bandwidths. In the late stages of reionization, identification of the largest ionized regions (which consist of foreground emission only) provides calibration points which potentially allow recovery of large-scale modes. Finally, we also show that (i) the broad frequency response of synchrotron and free–free emission will smear out any features in the electron momentum distribution and ensure spectrally smooth foregrounds and (ii) extragalactic radio recombination lines should be negligible foregrounds.