Can I switch the User-Agent header for only a single web site?

I have to use a particular web site for work, but this web site checks to see whether I'm using Internet Explorer and, if not, won't allow me to log in. I can switch my User-Agent header using the User Agent Switcher Firefox plugin and the web site works fine, but I really don't want to switch the header every time I go to the site and also don't want my User-Agent header to perpetually say I'm using IE.

I'm on Linux, so actually switching to IE unfortunately isn't an option.

Is there any way to define a particular User-Agent for a particular web site in either Firefox or Chrome?

The extension User-Agent Switcher for Chrome has a Permanent Spoof list, where you can set specific user agents for certain domains.

Configuration

1. Add the extension to Chrome.

2. Open chrome-extension://djflhoibgkdhkhhcedjiklpkjnoahfmg/options.html.

3. Click Permanent Spoof list.

4. Enter the domain of the particualr website, select the desired user agent and click Add.

Until Firefox 25, one could set general.useragent.override.[domain] to a custom UA string. Unfortunately, this feature was removed in Firefox 25.

Now, to get per-site UA string, one could install UAControl plus User-Agent JS Fixer. The first add-on modifies the User-Agent request header per site, but not the navigator.userAgent string that is often used for UA-sniffing. The second add-on complements the first add-on by overriding navigator.userAgent.

Since I already had Greasemonkey installed, and the site I'm targetting only uses client-side User-Agent sniffing, I decided to write a small user script to change navigator.userAgent for this particular site:

// ==UserScript==
// @name        Change navigator.userAgent
// @namespace   Rob W
// @description Changes navigator.userAgent to IE on IEGallery.com
// @match       http://www.iegallery.com/*
// @run-at      document-start
// @grant       none
// @version     1
// ==/UserScript==

Object.defineProperty(navigator, 'userAgent', {
    value: 'Mozilla/5.0 (compatible; MSIE 10.0; Windows NT 6.2; Trident/6.0)'
});

Works like a charm in Firefox 26.0.

For Firefox there are the extensions UAControl and ua-site-switch, which allow setting the User-Agent header on a per-site basis. Pick one of them and, as Rob W pointed out, you'll need to install User-Agent JS Fixer too.

There are many extensions you can use which can work by auto detecting the IE setting for the website on FF.

IE TabV2

The updated, fully supported IE Tab. Embed IE in a Firefox tab.
IE Tab Features: FF 4+ support, IE 7-9 compatibility modes, old IE Tab settings import. IE Tab supports ICBC and alipay

IE Tab Plus

The Original IE-Tab (Works also for IE8, IE9)

An enhanced version of IE Tab with Adblock Plus in IE supported. It also allows you import settings from IE Tab and synchronize cookies between IE and Firefox.

Only IE-Tab to support FF4 and beyond