I have a Shiny server installation, and a Git repo for my Shiny application. I develop my Shiny application and push it to the Git repo, and I would like the Shiny server to load the latest version of the application from the Git repo. How can I do this? I currently manually update the Shiny server folder for the application using a git pull every time that I push a new version of the application to the repo from my development machine, but I want to cut that step out of the process. 
You could use a git hook. Git hooks allow you to run a script whenever an action is completed. Inside your .git folder there's hooks folder with examples.
If you went this way, you'd probably want to use the post-receive hook which is triggered whenever you push to the repo and the server is done receiving the new commits.
You can find more information on git hooks here: http://git-scm.com/book/en/v2/Customizing-Git-Git-Hooks
Here's the part regarding post-receive hooks:
post-receive
The post-receive hook runs after the entire process is completed and can be used to update other services or notify users. It takes the same stdin data as the pre-receive hook. Examples include e-mailing a list, notifying a continuous integration server, or updating a ticket-tracking system – you can even parse the commit messages to see if any tickets need to be opened, modified, or closed. This script can’t stop the push process, but the client doesn’t disconnect until it has completed, so be careful if you try to do anything that may take a long time.
Here's an example:
#!/bin/bash
cd ~/webapps/site/ || exit
unset GIT_DIR
git pull
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