Using WebRequest, this will be the basic code
   var req = WebRequest.Create(@"http://irnafiarco.com/queue"))
   // prepare the request
   req.ContentType = "application/x-www-form-urlencoded";
   req.Method = "POST";
   // push the file contents into request body
   var data = "queue=" + System.IO.File.OpenText(filePath).ReadToEnd();
   var bytes = System.Text.Encoding.Ascii.GetBytes(data );  
    request.ContentLength = bytes.Length;
   var rs = req.GetRequestStream();
   rs.Write(bytes, 0, bytes.Length); 
   rs.Close ();
   // get the response
   var resp = req.GetResponse();
   var sr = new System.IO.StreamReader(resp.GetResponseStream());
   var result = sr.ReadToEnd();
Disclaimer: untested code
EDIT:
Added the post parameter name ("queue") which I have missed in first draft. Also added content-length for the request. This code should get you started. The basic idea is you need to simulate exact post request generated by PHP code. Use tool such as Fiddler/ Firebug on FF to inspect & compare request/response from PHP and .NET code.
Further, I suspect that the PHP code may generating request with content type as multipart/form-data. However,  I believe that server should also able to support the post body with application/x-www-form-urlencoded (because we have only one parameter in body) but in case it doesn't work and you must generate POST body as multipart/form-data then it will be little more involved. See this SO question where accepted answer has given the sample code for the same : Upload files with HTTPWebrequest (multipart/form-data)