How can I make a favicon stay static?

You can write a Greasemonkey script that uses a Javascript mutation observer to observe when the favicon changes and then change it back immediately before you notice anything.

Here is how to do this:

1. If you haven't done so already, install the Greasemonkey Firefox extension (this question is about Firefox, but you can also do this on Chrome and Edge by using the Tampermonkey extension: link for Chrome, link for Edge)

2. Click on the Greasemonkey menu in the top bar, then select "New user script..." (if you're using Tampermonkey, the button is called "Create a new script...")

   

3. When you've clicked on that button, it should open a new tab with a big text area where you can write Javascript code. The text area will probably already have some code in it. Simply delete that code and replace it with this:

   // ==UserScript==
   // @name         Do not change the favicon
   // @namespace    http://tampermonkey.net/
   // @version      0.1
   // @description  Prevents sites from changing the favicon
   // @author       You
   // @match        http://*/*
   // @match        https://*/*
   // @grant        none
   // ==/UserScript==
   
   (function() {
       'use strict';
   
       //Initialize some variables
       var favicon = document.querySelector("link[rel='shortcut icon']");
       const initialIcon = favicon.href;
   
       //Define the function to run when the favicon changes
       var observer = new MutationObserver(function(){
           if(favicon.href != initialIcon){
               favicon.href = initialIcon;    //Change the favicon back directly when it changes before you notice anything
           }
       });
   
       //Run the function above whenever the favicon changes
       observer.observe(favicon, {attributes: true});
   })();
   

   The lines on the top that start with // define properties of the script. @name defines the name of the script, and @match defines on which sites the script should be run (http://*/* and https://*/* means all sites).

   The part on the bottom is the actual script with keeps the favicon from changing. As I explained in the beginning of my answer, what it does is it detects when the favicon changes and quickly changes it back before you notice anything.

4. Press Ctrl+S to save the script, and then you should be done.

You'd have to edit the cookie tied to the site/web app and change the value that allows notification. There are extensions like CookieCutter that allow you to edit cookies although that one is for Safari.