<p>As the title says, is there a way to run the same Adobe AIR app more than once?  I have a little widget I wrote that shows thumbnails from a couple of photo streams, and I'd like to fix it so I can look at more than  one stream at a time.  Thanks!</p>

It seems that this is not possible. From the
documentation
:
Only one instance of an AIR application is started. When an already running application is invoked again, AIR dispatches a new invoke event to the running instance.
It also gives a possible workaround:
It is the responsibility of an AIR to respond to an invoke event and take the appropriate action (such as opening a new document window).
There is
already a bug
related to this on the bugtracker, but it is marked closed with no explicit resolution given...