Where to install Oracle client software on SharePoint 2010 Server
I do not know much about Sharepoint but in usual cases Oracle client has to be installed into server where application which connects to Oracle is located. So if your application will be running on Application server you should install Oracle client here.
Just for the information you can also try Instant Client. It uses less disk space and you do not have to install it. Just unpack where you prefer and point TNS_ADMIN environment variable to the directory where you unpacked it.
You can find variables by right clicking Computer icon then select Properties, Advanced System Settings, Environment Variables. And then add TNS_ADMIN to system variables.