To upgrade RHEL you need to have a subscription to Red Hat or convert the system to Oracle Linux.
Upgrading the OS does not directly affect Oracle ASM or the database, unless the upgrade breaks support of your system hardware. It is generally not an issue, but no information can be made based on your system description.
If you plan to switch to Oracle Linux with a ULN subscription, see https://linux.oracle.com/switch.html. In case of Oracle public yum, you can only upgrade to another release version without patches by setting up your yum configuration file accordingly, or upgrade to the latest release version including the latest errata and patches, which is 5.9.
If you pay for suppot with Red Hat, the must provide you the lastest repository to get the packages. But if you are using Linux without any support from Red Hat or Oracle, you can still upgrade your server.
Go to public-yum.oracle.com and follow the instructions in the page for your OS release, then as root do #yum update and thats all
There won't be changes in Oracle RDBMS behavior or complimentary steps