Disable the new mail system tray icon
From MozillaZine Knowledge Base
Thunderbird displays a new mail icon in the system tray even if you uncheck all of the checkboxs in Tools -> Options -> General -> "When a new message arrives".
Changing within Thunderbird
Versions earlier than 2.0
Supposedly you can disable that icon by setting the update.showSlidingNotification and mail.server.default.notify.on preferences to false using Tools -> Options -> Advanced -> General -> Config Editor. However, that doesn't seem to work with Thunderbird 1.5.
Open the configuration editor (Tools -> Options -> Advanced -> General -> Config Editor) and toggle mail.biff.show_tray_icon to false. (Windows only?)
Windows XP or later
A workaround is to configure Windows to hide the icon for you. The system tray doesn't have to show the icon when you do this.
Earlier versions of Windows
Use a utility program such as TraySaver to hide the icon for an application in the system tray.
You need may to install the Mozilla New Mail Icon (Biff) extension to display an icon in the standard (FreeDesktop.org) system tray (used by GNOME, KDE and IceWM) when new mail arrives. Uninstall it if you don't want this functionality. There is a seven year old bug report requesting new mail notification in the system tray. See updating add-ons if it doesn't support your version of Thunderbird for how to bypass the version checking. Frequently if you do that an older add-on will work.
Ubuntu installs Thunderbird without that extension, and displays an icon in the system tray when new mail arrives. It installs a Alltray program thats designed to "dock any application with no native tray icon (like Evolution, Thunderbird, Terminals) into the system tray.". They also have builds for SuSE, Fedora, Debian , Gentoo and "generic". Try searching for tray and/or Thunderbird in your distributions package manager to see if you have a similar package.
There is also the FireTray add-on. It supports Firefox, Thunderbird, Sunbird, Seamonkey 2 and Songbird under Linux, letting you use a custom icon, hide to tray instead of closing, and display the number of unread messages .