Special thanks to Daniel Dad for finding this solution

Normally the Finder is stuck in the dock, even if the Finder has been quit. A special setting can be added to one of the dock's configuration files to allow the Finder icon to be removed:

1. Navigate to /System/Library/CoreServices/Dock.app/Contents/Resources/ and open DockMenus.plist with Property List Editor
2. Click on the item called finder-quit and press the “Add Item” button in the toolbar
3. Click on the newly created item named Item 1 and change its type to Dictionary
4. Press the “Add Child” button two times to create two new items inside of Item 1
5. Change the type of the first new item to Number. Change its name to command and its value to 1004
6. Change the name of the second item to name and value to REMOVE_FROM_DOCK
7. Save the file and enter your administrator password when prompted
8. Open Activity Monitor (located in /Applications/Utilities), select the Dock from the list of active processes, and force quit it. The dock will relaunch automatically after a few seconds.
9. In Path Finder, select the menu Path Finder > Quit Finder
10. Right-click or control-click on the Finder's dock icon and select “Remove from Dock”

The Finder can be launched again at any time by opening it from /System/Library/CoreServices. For quick access, drag the Finder application from the CoreServices folder to Path Finder's sidebar or toolbar.

After successfully removing the Finder from the dock you may wish to still use it. There are a handful of Finder features that Path Finder does not yet support, such as Time Machine's restore browsing. Time Machine Backup will still work properly without Finder, since it is done by a system process different than Finder.

The following procedure will allow the Finder to run silently in the background. It will not display an icon in either the dock or the command-tab application switcher. Time Machine Restore should work normally, as should all other apps that require use of the Finder:

1. Quit the Finder
2. Navigate to /System/Library/CoreServices/Finder.app/Contents/ and open Info.plist in either a text editor or Property List Editor (only available if you have Apple's Developer Tools installed). You will need root permissions to edit this file, so you may wish to use Commands > Launch as Root in Path Finder.
   1. If you're using a text editor find the line <key>LSUIElement</key>. Right below it change <false/> or <string>0</string> (depending on which occurs) to <true/>. If these lines do not exist, add them. Save the file.
   2. If you are using Property List Editor, find the line “Application is agent (UIElement)” and mark its check box. If the line does not exist, add it using the “Add Item” button in the toolbar.
3. Relaunch the Finder from the CoreServices folder
4. Remove the Finder icon from the dock using the method outlined in the first post

If you would still like to access the Finder directly, you can drag the app from the CoreServices folder into your Path Finder sidebar, toolbar, or shelf.