Only if you want to do something that would otherwise be a violation of copyright - e.g. distribute the module to other people (assuming it's sufficiently entangled with linux to be a derivative work therof). The GPL only licenses you to do things that you otherwise couldn't do, it doesn't restrict you from doing things that were never a violation of copyright (e.g. privately modifying your own things).

No kernel modules can be closed source; GPU drivers are a common example of a closed source kernel module.