Here is my official WordPress forum support plea: http://wordpress.org/support/topic/headerphp-is-not-updating?replies=4
I changed all the permissions via FTP, currently set to 777. Originally 666, but that didn't work, so here we are.
I cleared all of my browser caches.
I added the following line to wp-config.php:
define('DISABLE_CACHE', true);
I don't know what else to do at this point. The file on my server reflects the changes I made, but if you look at the source on the website, via Firebug or whatever, it isn't there.
From the native WordPress editor, it says I can't edit the file because the permissions aren't set. What's really absurd is that this is a custom theme specifically for this website. Whose business is it if I want to change it?
Can anyone please at least verify that the correct 'header.php' is in wp-content/themes/themeofchoice?
EDIT: I don't know what happened but suddenly my changes have taken effect. I guess Wordpress takes hours to update .php files? WOW.
Thank you for your answers everybody. If anyone has similar problems, I guess take the same steps I did. If anything changed, I continued to change the permissions of all the parent folders as was suggested.