I get notifications of old IMAP emails, but at most they're only a few days old. It's inconsistent, but I don't think a day goes by without seeing at least one meaningless notification.
It sometimes is triggered by "reply" -- let's say an email comes in, there's a notification. I go to Mac Mail and reply. When my reply goes out, I get a notification that the email I've just replied to has come in.
My email is hosted at 1and1.com. I haven't gone there and checked all the settings. I have several accounts there and I'll start checking the bad notifications to see if they all come from one (perhaps misconfigured) account, or from all of them.
Frankly, Mac Mail under Mavericks has been so consistently weird for me that I've just been shrugging my shoulders and blaming it rather than considering whether my mail server might be at least partly at fault. Each time there's a point upgrade I hope for better Mail, but it hasn't happened.