well i couldn't care less about firefox and use it very minimally but you can do this with IE too. IE you can't drag the button to your links but you can copy and paste it. when you click on the button it will take you to the Tiny URL page but it will have already created the URL and pasted it into your clipboard. I didn't realize it at first and didn't trust it at first when i did realize it, but it does work.
The only thing with that is that i have notice the beta version of Norton doesn't allow the site access to my clipboard so i still have to copy and paste it. I use it all the time at home though.... mostly for my twittering