I found a way to get Tasker to clear another app's notification from the status bar. In general the Android OS doesn't directly allow an app to mess with another app, as it would be a security issue.
However, the trick is that most apps will clear their own notifications once you launch them. The
example I use here is clearing the new SMS notification from the Messaging app.
Create a Tasker task with these steps://
+ > App > Load App > Messaging
+ > Task > Wait > 2s
+ > App > Go Home