To be more precise, in the case where a player downloads Diablo III or is patching Diablo III and starts playing it as soon as it becomes "Playable", but well before it hits "Optimal", does Diablo III continue to download patches during gameplay?
No it's already finished downloading everything. This is from their website:
This process removes unnecessary files and consolidates data where possible. The process requires 2-3 GB of available hard drive space but your game folder will be smaller once it is complete.
Obviously it doesn't always require that much space/consolidate that much some patches don't optimize anything.