It appears that Instagram is not auto-unsubscribing from high volume tags UNLESS your server is not responding quickly enough.
Although it was not obvious to me at the time, it seems that there must have been times where my server was taking 2 seconds to respond, possibly due to concurrent calls on some blocking resource.
After posting this question, I ripped out everything from my HTTP responder except for one method call to copy the data that Instagram sent; everything else happens asynchronously, so if there is any blocking it does not prevent the HTTP response from going back to Instagram. I am able to stay subscribed to multiple high-volume tags (including the #1 tag on Instagram) without getting auto-unsubscribed.
It also does seem to be true (and undocumented) that if you fall behind/respond slowly, you get unsubscribed from the problematic tag.