Why does the Outlook taskbar envelope icon sometimes remain active, even if I have no unread email?

This is a known bug in Outlook. To get rid of this icon just right-click any Outlook folder with all read mail select Mark All as Read in the context menu.

Another way of getting rid of the icon is to fully open an email - any email - in a new window (not just in the preview pane) by double-clicking it.

For those who likes keyboard shortcuts, there is a faster workaround than the right click/mark all as read. Select any message in your Inbox and use CTRL+U, CTRL+K. This will mark the message as Unread, then Read, and the envelope will vanish.