No, but the url must be a direct link to the image. To get this, go to the photobucket page with your image, right click the image and select "copy image location" then just paste it between the img tags.
I figured it out. For the benefit of anyone else who reads this, let me explain the problem.
When you click on "edit signature", you get a box to paste your link into. I did that correctly, with the tags and all. But, underneath are 3 checkboxes that needed to be unchecked. It was probably the one about parsing URLs that was stopping me.