Godot requires a tool called rcedit to change the application icon. According to my understanding of GPL, I believe I have to include the source code of Godot with the releases of my game. Do I also need to include the source code for rcedit?
As far as I can tell, Godot is licensed under the MIT license which only requires you to preserve and reproduce a copyright notice and simple license terms. You do not need to share any of your source (or Godot's source) whatsoever, if you don't want to.
Even if Godot were under the GPL, and you did need to share the source code of your application built on Godot code, you would not need to include the source code of the rcedit tool. Admittedly, rcedit (or another tool for modifying Windows executables) is needed to produce the final binary as-is, but the GPLv3 excuses it from being part of the required source in the following section:
The “Corresponding Source” for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include [...] general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work.
Insofar as rcedit is a "general purpose tool" or is a "generally available free program[...] used unmodified" (in fact, it appears to be both) that is not part of the work, you do not need to distribute it along with your program's source.