I have Thunderbird 9.0.1 and iOS MobileMail.app (iPhone 4S) connected to the same IMAP account on a Dovecot server.
When I delete emails from Thunderbird, they no longer show up on the iPhone. So far so good.
When I delete emails from the iPhone, the emails still show up in Thunderbird. I've tried waiting up to an hour after I've deleted the emails from my iPhone, but the emails I deleted still show up in Thunderbird. I've tried clicking the "Get Mail" button, but deleted emails still show up in Thunderbird. The only remedy I've found so far is to quit and restart Thunderbird, which is inconvenient.
Up until just recently, I used Apple's desktop Mail.app, and emails deleted on the iPhone disappeared immediately from desktop Mail.app.