As of Firefox 57, this is not possible from an extension.
WebExtensions do not permit the window icon to be changed from an extension.
Prior to Firefox 57 (or non-release versions w/ legacy add-ons)
The combination of the title of your question and the text of your question make it unclear what you desire to accomplish.
If your goal is to dynamically change the window icon of a currently running Firefox process then you will need to follow something along the lines of the second or third method listed in nmaier's answer.
If you goal is to always have a different, static icon used for the primary Firefox windows for a specific profile, that is quite easy.
You will need icon files of the appropriate format for each architecture for which you desire this to work.
The following assumes Windows, it is easily expanded to other architectures by including an icon file with the same name, but appropriate file extension and format.
Create a simple overlay, extracted extension. You will need a minimum of 2 files:
- <extension-dir>instal.rdf
- <extension-dir>\chrome\icons\default\main-window.ico
Example, fully functional, install.rdf:
<?xml version="1.0"?>
<RDF xmlns="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
xmlns:em="http://www.mozilla.org/2004/em-rdf#">
<Description about="urn:mozilla:install-manifest">
<em:id>[email protected]</em:id>
<em:version>1.0.0</em:version>
<em:name>Window icon change</em:name>
<em:description>Change the Firefox main window icon.</em:description>
<em:creator>Makyen</em:creator>
<em:unpack>true</em:unpack>
<em:targetApplication>
<Description>
<em:id>{ec8030f7-c20a-464f-9b0e-13a3a9e97384}</em:id>
<em:minVersion>3.0a1</em:minVersion>
<em:maxVersion>43.0a1</em:maxVersion>
</Description>
</em:targetApplication>
</Description>
</RDF>
NOTE: The "<extension-dir>\chrome\icons\default\" directory is different than the one in the response by nmaier. In that answer the "icons" and "default" directories reversed and will be non-functional.
You will probably want an additional file:
<extension-dir>chrome.manifest
The chrome.manifest file is not required. However, not having it may result in a single line being printed to the error/browser console (if you even have that open). If the chrome.manifest file exists, even if zero length, there will be no complaint in the console that the file could not be read.
Install the extension. The easy way to do this is to create a zip file with those three files; then change the file extension to .xpi; then drag and drop it onto a Firefox window running the profile in which you desire it to be installed.
You can expand this to include icons for whatever sub-windows you desire. You will need to determine the ID for each sub-window. The icon file name is just the window ID with the appropriate extension for an icon in the architectures you desire. "main-window" is just the ID for the main Firefox browser window.
Creating an extension to test this took less than 5 minutes. You should find it reasonably easy to accomplish.
This assumes that there is not a custom main-window icon located at (Windows, default install location):
C:\Program Files\Mozilla Firefox\browser\chrome\icons\default
as that directory is for all profiles and is searched first.
This will not work if the extension is either restartless or extractionless.
You can find a brief amount of information about window icons on MDN. nmaier mentions the docs talking about bundles. When talking about Mozilla add-ons, a bundle is your add-on package.
The icon file(s) will be at (Windows):
<profile dir>\extensions\<extension-dir>\chrome\icons\default*
Once the extension is installed, you can change it/them manually without re-installing, if desired.
Add-ons created to solve this
Based on the discussion in the comments, I created a Firefox Add-on to allow setting the window icons for the profile. It is much expanded upon the 5 minute add-on mentioned in the comments. The addition is entirely in a UI for the options dialog for selecting the icon to use and assigning it to the various different windows Firefox opens. You can get it from Mozilla Add-ons under Change Profile's Window Icons. Unfortunately, it's not possible for that add-on to function as of Firefox 48 which requires add-ons to be signed. To dynamically change the icon requires changing files which must be signed. Thus it's not possible to dynamically change the icon with add-on signing required.
Instead I created a few add-ons which statically change the window icon. You can find them on AMO.