Here's a userscript that hides the red indicator. Tested with ViolentMonkey in Firefox. Should work in Chrome.

Maybe a bit more complex than it needs to be but I'm unsure if the indicator is updated live (i.e. after the page is loaded) so reused the mutation observer template I had...

    // ==UserScript==
    // @name          SEHideReviewIndicator
    // @namespace     http://vulpin.com/
    // @description	  Hide the red indicator on review queues
    // @match         *://*.askubuntu.com/*
    // @match         *://*.mathoverflow.net/*
    // @match         *://*.onstartups.com/*
    // @match         *://*.serverfault.com/*
    // @match         *://*.stackapps.com/*
    // @match         *://*.stackexchange.com/*
    // @match         *://*.stackoverflow.com/*
    // @match         *://*.superuser.com/*
    // @version       1.0.0
    // @grant         none
    // ==/UserScript==
    
    (function() {
    var Program = {
        main: function() {
            window.removeEventListener('load', mainEventListener, false);
            
            var node = document.querySelector('.js-review-button');
            
            var observer = new MutationObserver(function(mutations, observer) {
                mutations.forEach(function(mutation) {
                    Program.processNode(mutation.target);
                });
            });
            
            observer.observe(node, {
                attributes: true,
                attributeFilter: ['class']
            });
            
            Program.processNode(node);
        },
        
        processNode: function(node) {
            if (node.classList.contains('_danger-indicator')) {
                console.log('Hiding red indicator on review button ', node);
                node.classList.remove('_danger-indicator');
            }
        }
    };
    
    var mainEventListener = Program.main.bind(Program);
    
    window.addEventListener('load', mainEventListener, false);
    })();

Also doesn't really handle recursion so it could in theory get stuck in a loop, but in practice that would only happen if the indicator were re-added every time I remove it. Which would be an odd action for the site scripts to take.