How to automate printing of multiple parts continuously?
The only thing I can think of off hand is an old mod for the early MakerBot machines. It first was released for the Thing-O'-Matic I believe, but is compatible with Replicator 1 machines (and its knock-offs). Here's the Thingiverse page, but look up Automatic Build Plate.

Essentially, you can use the Replicator G slicing program and there is a setting for "ABP" or Automatic Build Plate. This will basically tell the ABP to run its routine after the controller receives the response that the printing program is done and roll the finished part off the edge of the build plate, then start the same program over again.

Drawbacks:


I don't think it's easily compatible with newer machines/slicers. But, it's open source
Pretty sure you have to use Replicator G, which is outdated now and may make your machine sound like it's going to fall apart (I know from experience)


Going off of @Pete's answer about solenoids. It reminded me that someone integrated a solenoid "ejector" (aka Boxing Glove) for their machine.

Update (06/08/2016):

Forgot to mention that if you choose to create your own "Boxing Glove" or conveyor belt, some software such as Octo-Pi and Repetier-Host allow plugins. So, you could interface with your hardware via customized code and integrate the functionality directly into the slicing application for the full closed loop operation.