Optoacoustic tomography (OAT), also known as photoacoustic tomography, is a rapidly emerging hybrid imaging technique that possesses great potential for a wide range of biomedical imaging applications. In OAT, a laser is employed to illuminate the tissue of interest and acoustic signals are produced via the photoacoustic effect. From these data, an estimate of the distribution of the absorbed optical energy density within the tissue is reconstructed, referred to as the object function. This quantity is defined, in part, by the distribution of light fluence within the tissue that is established by the laser source. When performing three-dimensional imaging of large objects, such as a female human breast, it can be difficult to achieve a relatively uniform coverage of light fluence within the volume of interest when the position of the laser source is fixed. To circumvent this, researchers have proposed illumination schemes in which the relative position of the laser source and ultrasound probe is fixed, and both are rotated together to acquire a tomographic dataset. A problem with this rotating-illumination scheme is that the tomographic data are inconsistent; namely, the acoustic data recorded at each tomographic view angle (i.e., probe position) are produced by a distinct object function. In this work, the impact of this data inconsistency on image reconstruction accuracy is investigated systematically. This is accomplished by use of computer-simulation studies and application of mathematical results from the theory of microlocal analysis. These studies specify the set of image discontinuities that can be stably reconstructed with a nonstationary optical illumination setup. The study also includes a comparison of the ability of iterative and analytic image reconstruction methods to mitigate artifacts attributable to the data inconsistency.