I replied via email, but I wanted to post the answer here in case anyone else had the same question.
The Drop Shadow effect had been applied to the object state as opposed to the contents of the object state. When you select the contents of the object state and apply the Drop Shadow, then it appears when it's exported.
You can see the difference between what is selected in the Objects States panel. When the object state is selected, InDesign displays a bounding box icon with no fill. When the contents of the object state are selected, then the bounding box icon has a fill.
Just as an aside--the same is also true of animation applied to the contents of a multi-state object (MSO). The animation needs to be applied to the contents of the object state and not the object state itself.