You could certainly have your desktop application communicate "live" with the server. There is actually a reasonably easy way to host .NET remoting in IIS. The ASP.NET web app can certainly communicate with that as well. With the right tricks you can get the desktop application subscribed to events at the server so that the server can raise events the desktop client responds to. However, because of the stateless/passive nature of a web application it will be a bit more difficult to have a web app observe server events. If fact, you have to make the web app monitor the server using something like polling if you want automatic response to server side activity.
I have actually been playing with some code Tomer Gabel to make two-way remoting work: http://www.tomergabel.com/Events+And+NET+Remoting.aspx