HOWTO transfer a request made to an Apache server to an IIS server without using a URL redirect?

If I understand your situation correctly, I believe you may be able to use mod_proxy to do what you want. Basically, Apache would act as a "reverse proxy" for the requests that you actually want to serve from IIS. Note that a reverse proxy isn't like a normal HTTP proxy. From the mod_proxy docs:

A reverse proxy ... appears to the client just like an ordinary web server. No special configuration on the client is necessary. The client makes ordinary requests for content in the name-space of the reverse proxy. The reverse proxy then decides where to send those requests, and returns the content as if it was itself the origin.

Yes, what Laurence Gonsalves says. Here is a simple as simple can be configuration file that I've taken from a caching reverse proxy server of mine. You should be able to work in some URL matching but I've not done that myself.

# httpd *reverse proxy caching server* config file for apache httpd 2.2ServerRoot "C:/Program Files/Apache Group/Apache2"#must exist, no reason to have anythign in it    DocumentRoot "C:/Program Files/Apache Group/Apache2/htdocs"Listen 127.0.0.1:80Listen 192.168.1.33:80ServerAdmin stu@thompson.nameServerName proxy.server.comLoadModule auth_module modules/mod_auth.soLoadModule log_config_module modules/mod_log_config.soLoadModule mime_module modules/mod_mime.soLoadModule proxy_module modules/mod_proxy.soLoadModule proxy_http_module modules/mod_proxy_http.so### PROXY CONFIGURATION  ###ProxyRequests OffProxyVia On### VH ###<VirtualHost 192.168.1.33>    ProxyPass        /  http://192.168.1.34:80/    ProxyPassReverse /  http://192.168.1.34:80/    LogLevel info</VirtualHost>### LOGGING CONFIGURATION #### error log will not catch proxied contentErrorLog logs/error.logLogLevel infoLogFormat "%{Host}i %v %h %l %u %t \"%r\" %>s %b  commonCustomLog logs/access.log commonTypesConfig conf/mime.types