Here is a screenshot showing the issue:
When I first loaded the imagery into the mosaic and built overviews, there was a massive black box around the entire dataset. I got rid of that by going into the mosaic dataset properties and changing the NoData Values to 0,0,0. The solid black border disappeared, but this area remains. It only shows at overviews zoomed out beyond 1:52k. Once you zoom in, they disappear completely, so it seems to be a problem specific to the higher level overviews.