This took me forever to figure out, but I think I have your answer. When you play back multicam in premiere, it tries to mix all of the audio from your original sequence. Even though all you hear is the audio from track 1 (at least this was the behavior in previous versions).
My fix for this was to delete all all of the audio clips (besides track 1) in the original nested sequence. Since I use external audio, I just use the camera audio for syncing purposes anyway. But that's pretty much what you'll want to do. Mix down whatever audio you plan on actually using for your project and put that in the sequence with your multicam clip. Or if you prefer, put it on track 1 of your nested sequence.
Also, you'll want to unlink the audio and video from your multicam clip, and delete any cuts you've made to the audio. For some reason, edits to the audio clip also cause the "rendering required files" error. So just delete all of the many audio "clips" you've made throughout the process of editiing your video, saving just one to drag out to the end of your video. This way you'll only have one long audio clip spanning the entire length of your video. Bam. No more rendering required files.
MomkeyEatsMonkey, you nailed it. No more rendering required files errors. Thanks!