I'm using Firefox 7.0.1 and I have no difficulty at all with using the short codes in the support docs. That makes me think you may have an issue that troubleshooting may clear up.
How are you connecting to the internet and to WordPress.com? The way you connect to the internet (mobile, satellite, DSL, dial-up) and to your blog, and how many proxy server jumps etc. it takes to connect can cause problems. There maybe be problems with your ISP and/or with the proxy servers.
Are you using http:// or https:// see http://en.support.wordpress.com/https/
Which browser and version of it are you using? Have your tried clearing your browser cache and cookies? http://en.support.wordpress.com/browser-issues/#clear-your-cache-and-cookies
Is your browser version up to date? You can upgrade any browser version here.
Do you have third party cookies enabled? Do you have another browser you can try?
You can use this link to see if your Flash is up to date. http://www.adobe.com/software/flash/about/ If it’s not up to date then this is where to download and install the latest version of Flash. http://www.adobe.com/products/flashplayer/