Thunderbird Conversations 2.2.1 has been released.

Open this PDF in a new tab

You can now view PDF attachments inside Thunderbird using pdf.js! You don't need an external viewer anymore. That's the pdf.js magic.

PDF Viewer

Clicking the icon opens a new tab with a mini-viewer (very basic functionality). It works pretty well, and the plan is to add a viewer inside the conversation, so that you can get some sort of inline pdf viewer, just below the email. That's a reasonably sized chunk of work, but I'm really busy right now, so it's not going to happen anytime soon; therefore, anyone coming with a patch would be extremely well-received :).