Outlook must be open for Reminders or alerts to fire. You MAY elect to
"hide when minimized" by r-clicking the Outlook Icon in the notification
area and enabling that option.

Alan B wrote:

I am having this same problem. My Outlook alerts only appear if I have an
Office application up.
I have a colleague who doesn't have this problem. I noticed that my taskbar
looks different than his. Mine has the newer Windows 7 look with square
icons, while his looks more like the traditional Windows XP look with icon
and name of the application next to it. I don't know if this is related or
not.

I've also found that my IM alerts aren't as noticeable in Windows 7.