Are there any apps that copy files to another location whenever they change? I've been trying to change my IDE from NetBeans to Sublime Text.

The most important thing that keeps me using NetBeans is the "Copy files from Sources Folder to another location" feature.

I can't find a plug-in for Sublime Text that can perform this task.
Your problem can be solved using 'Folder Actions'. You can create folder actions using AppleScript or Automator. If you do not know what 'Folder Actions are, you can read the Folder Actions Reference page on the Apple site to get you started.


  Folder Actions Reference
  
  Folder Actions is a feature of OS X that lets you associate
  AppleScript scripts with folders. A Folder Action script is executed
  when the folder to which it is attached is opened or closed, moved or
  resized, or has items added or removed. The script provides a handler
  that matches the appropriate format for the action, as described in
  this chapter.
  
  Folder Actions make it easy to create hot folders that respond to
  external actions to trigger a workflow. For example, you can use a
  Folder Action script to initiate automated processing of any photo
  dropped in a targeted folder. A well written Folder Action script
  leaves the hot folder empty. This avoids repeated application of the
  action to the same files, and allows Folder Actions to perform more
  efficiently.
  
  You can Control-click a folder to access some Folder Action features
  with the contextual menu in the Finder. Or you can use the Folder
  Actions Setup application, located in /Applications/AppleScript.


Note: you can create your own 'Folder Actions' using AppleScript Editor or Automator.