This is driving the whole team crazy. There must be some simple mis-configured part of IIS or our Web Server, but every time we try to run out ASP.NET Web Application on IIS
I had these exact symptoms and my issue was similar to Peter's. Was setting up an existing project on a new server. My project referenced the IIS7 URL Rewriting module, but that hadn't been installed yet on the new server. Installing it fixed my issue.
You can use the Microsoft Web Platform Installer to install it. Execute it, select Products, in the left menu select Server and find URL Rewrite in the list and install it.
Or you can download it here.