I installed it on a multisite. That would be the most significant difference from my other installs.
I installed the plugin, activated the plugin, activated with the license key, installed IDE and then set a pin. I received my first puzzled phone call from a colleague shortly after.
If I wasn't logged in, it would redirect visitors to the IDE login (as like on the Instant IDE Manager Dashboard). An image is below, as seen by my colleague. This forced login was on the front-end for all visitors.
When I was logged-in to Wordpress it would display my homepage on the front-end (though I think this might have been just a cached page) and every other page was a 404 error. I didn't try logging-in on a fresh browser without cached files to test this out further. I didn't do much testing because I was in a bit of a rush to fix it. I could still access the Wordpress dashboard fine. I tried to uninstall IDE, deactivate and uninstall the the plugin. But it made no difference so I just restored directly from a back up.
I've tried to recreate the problem again on a fresh multisite, but the same problem didn't occur. Maybe it was something unique to my particular setup. I'll keep tinkering.