I’m trying to alter the header.php and footer.php but after saving my changes, the alterations are not showing up on the live site.
Is there anything I need to do to push these changes live?
Nope, changes are reflected immediately. Even browser caching is usually not a problem for those kinds of changes because the PHP-generated content generally isn’t cached. You could have some server-side caching setup going on but that’s usually not the case.