It can, if you want it to do so. You will want to modify your routing tables so that the traffic for any specific content is routed through the OpenVPN connection. This can be done via --push "route" directives on the OpenVPN server. There are also --redirect-gateway and --redirect-private directives which are experimental "forward everything" commands.
Otherwise, the specifics of how really depends on your OS, you need the proper routes set, and for example in Windows you need to set your virtual interface at a higher priority than your other interfaces via Advanced adapter settings - there should be information specifically how to do this for your situation.
Check out:
http://openvpn.net/index.php/open-source/documentation/howto.html#redirect