Remove “Thank you for your trust in WordPress” from the bottom of the dashboard

The following code can be used to remove the text from the dashboard:

function cwhf_remove_wp_footer_admin()
{
    add_filter( 'admin_footer_text',    '__return_false', 11 );
}
add_action( 'admin_init', 'cwhf_remove_wp_footer_admin' );

Instead of an empty text, you can also set your own text here.

Remove WordPress version from the bottom of the dashboard

The version can be removed in this way:

function cwhf_remove_version_wp_footer_admin()
{
    add_filter( 'update_footer',        '__return_false', 11 );
}
add_action( 'admin_init', 'cwhf_remove_version_wp_footer_admin' );

To hide both, you should ideally combine both filter calls:

function cwhf_remove_wp_footer_admin()
{
    add_filter( 'admin_footer_text',    '__return_false', 11 );
    add_filter( 'update_footer',        '__return_false', 11 );
}
add_action( 'admin_init', 'cwhf_remove_wp_footer_admin' );