I agree with Joyce in that there are multiple ways you could go about this. IIS7 does in fact support PHP and, for the most part, it works fairly well. With the WordPress site running in IIS, it is pretty straight forward to create a virtual directory (IIS application) to run your ASP.Net application on the same domain as your main site (ex. http://mydomain.com/ASPdotNetApp/).
If you want to keep WordPress running on your Linux server, then there are a couple of options that are probably the most common. The first would be to run your ASP.Net application on a separate Windows server as a subdomain. So your main site would still be http://mydomain.com and the new application would be http://aspnetapp.mydomain.com.
The other common option for running the ASP.Net application on a different server would be to use a reverse proxy. This was alluded to in a couple of the answers given so far but I don't think it was explicitly stated. To do this, you would run your ASP.Net application on a different server and then setup mod_proxy on your Linux server (specifically, by using the ProxyPass directive).
Information on mod_proxy can be found here and information on ProxyPass can be found on that page here.
With the reverse proxy in place, if you were to direct users to http://mydomain.com/AspdotNetApp/, they would not be aware that the application was running on a different server.