If the orientations of galaxies are correlated with large-scale structure, then anisotropic selection effects such as preferential selection of face-on disc galaxies can contaminate large-scale structure observables. Here we consider the effect on the galaxy bispectrum, which has attracted interest as a way to break the degeneracy between galaxy bias and the amplitude of matter fluctuations σ8. We consider two models of intrinsic galaxy alignments: one where the probability distribution for the galaxy's orientation contains a term linear in the local tidal field, appropriate for elliptical galaxies; and one with a term quadratic in the local tidal field, which may be applicable to disc galaxies. We compute the correction to the redshift space bispectrum in the quasi-linear regime, and then focus on its effects on parameter constraints from the transverse bispectrum, i.e. using triangles in the plane of the sky. We show that in the linear alignment model, intrinsic alignments result in an error in the galaxy bias parameters, but do not affect the inferred value of σ8. In contrast, the quadratic alignment model results in a systematic error in both the bias parameters and σ8. However, the quadratic alignment effect has a unique configuration dependence that should enable it to be removed in upcoming surveys.