Per this Trac ticket, WP intentionally disallows adding the ‘unfiltered_html’ capability to any non-Super Admin users on the WordPress multisite network. This is due to potential security vulnerabilities that could result from doing to. As that ticket points out:
If you just want those users to be able to insert things like YouTube video iframes, you could instead use WP’s built-in embed shortcode to embed that content instead.
If that’s not enough, and you need to extend the ‘unfiltered_html’ capability to non-Super Admins, the code below can be used to do that. Just be sure that you trust those users 100% – with great power comes great responsibility.
Just change ‘editor’ on line 13 with whichever user role you need to add the unfiltered_html capability to.
Props to Justin Tadlock for the code he posted on this thread.
Also see this plugin by Automattic that gives both Administrators and Editors the unfiltered_html capability on multisite installs: https://wordpress.org/plugins-wp/unfiltered-mu/