One well-known way to constrain the hydrogen neutral fraction, x_H, of the high-redshift intergalactic medium (IGM) is through the shape of the red damping wing of the Lya absorption line. We examine this methods effectiveness in light of recent models showing that the IGM neutral fraction is highly inhomogeneous on large scales during reionization. Using both analytic models and semi-numeric simulations, we show that the picket-fence absorption typical in reionization models introduces both scatter and a systematic bias to the measurement of x_H. In particular, we show that simple fits to the damping wing tend to overestimate the true neutral fraction in a partially ionized universe, with a fractional error of ~ 30% near the middle of reionization. This bias is generic to any inhomogeneous model. However, the bias is reduced and can even underestimate x_H if the observational sample only probes a subset of the entire halo population, such as quasars with large HII regions. We also find that the damping wing absorption profile is generally steeper than one would naively expect in a homogeneously ionized universe. The profile steepens and the sightline-to-sightline scatter increases as reionization progresses. Of course, the bias and scatter also depend on x_H and so can, at least in principle, be used to constrain it. Damping wing constraints must therefore be interpreted by comparison to theoretical models of inhomogeneous reionization.
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