<p>Like the title says, If I place an <code>app_offline.htm</code> in the application root, will it cut off currently running requests, or just new ones?</p>

From ScottGu's blog:
Basically, if you place a file with
  this name in the root of a web
  application directory,
ASP.NET 2.0
  will shut-down the application, unload
  the application domain from the
  server, and stop processing any new
  incoming requests for that
  application
.  ASP.NET will also then
  respond to all requests for dynamic
  pages in the application by sending
  back the content of the
  app_offline.htm file (for example: you
  might want to have a “site under
  construction” or “down for
  maintenance” message).
App_Offline.htm - ScottGu's Blog
So, it seems like it will continue processing current request, but stop new incoming requests.