Thank you for the suggestion, Dan! Unfortunately, that did not work for me. I created a geodatabase, converted the shapefile to a geodatabase using "Feature Class to Geodatabase", then tried running the Polygon Neighbors tool on the Geodatabase file. The good news is that it didn't give me a geometry error; but it still gave an error (a pretty uninformative one, basically that it failed to execute). I also tried the "Feature Class to Feature Class" conversion after creating a Geodatabase, but that gave a geometry warning as well.
Fortunately, I found a workaround. I just converted the polygons to lines and checked the box that reports bordering FIDs, which is all I really needed.