Disable user profile editing for one user

If you want to completerly remove the possibility for an user to access on profile, you can

1. Give admin possibility to choose which user can access on profile and which not
2. Remove the profile link from menu and from admin bar
3. Ensure the profile is not accessible even if accessed directly by manually typing address

Step 1

First point is pretty easy using ‘personal_options’, ‘edit_user_profile_update’ hooks:

add_action( 'personal_options', 'rpa_profile_ban_field' );
add_action( 'edit_user_profile_update', 'rpa_profile_ban_field_save' );

function rpa_profile_ban_field( \WP_User $user ) {
  $current = wp_get_current_user();
  if ( ! is_admin() || $user->ID === $current->ID ) return;
  if ( ! user_can( $current, 'edit_users' ) ) return;
  $target = new WP_User( $user->ID ); 
  if ( $target->exists() &&  ! user_can( $target, 'edit_users' ) ) {
    $banned = (int) get_user_meta( $user->ID, '_profile_banned', TRUE );
  ?>
  <table class="form-table"><tbody><tr>
  <th scope="row">Profile Ban</th><td>
  <input<?php checked(1, $banned); ?> name="_profile_banned" value="1" type="checkbox">
  Ban user to enter profile?
  </td></tr></tbody></table>
  <?php
  }
}

function rpa_profile_ban_field_save( $userid ) {
  $current = wp_get_current_user();
  if ( ! is_admin() || $user->ID === $current->ID ) return;
  if ( ! user_can( $current, 'edit_users' ) ) return;
  $target = new WP_User( $userid );
  if ( ! $target->exists() || user_can( $target, 'edit_users' ) ) return; 
  $ban = filter_input( INPUT_POST, '_profile_banned', FILTER_SANITIZE_NUMBER_INT );
  if ( (int) $ban > 0 ) {
    update_user_meta( $userid, '_profile_banned', 1 );
  } elseif ( get_user_meta( $userid, '_profile_banned', TRUE ) ) {
    delete_user_meta( $userid, '_profile_banned' );
  }
}

Using this code administrators can go to each user and disable the access to profile page.
This is the output of the code:

Step 2

Remove profile link from menu is easy using remove_menu_page:

add_action( 'admin_menu', 'rpa_profile_menu_remove' );

function rpa_profile_menu_remove(){
  $remove = get_user_meta( get_current_user_id(), '_profile_banned', TRUE );
  if ( ! current_user_can( 'edit_users' ) && (int) $remove > 0 ) {
    remove_menu_page( 'profile.php' );
  } 
}

Remove profile link from admin bar is a little more complex, or better, completely remove the user menu on top right is easy using $wp_admin_bar->remove_menu('my-account') (see WP_Admin_Bar docs), but it is “unaesthetic” and also you remove the possibility to logout.
So a better solution is remove just the link to profile:

add_action( 'wp_before_admin_bar_render', 'rpa_profile_adminbar_remove' );

function rpa_profile_adminbar_remove() {
  $remove = get_user_meta( get_current_user_id(), '_profile_banned', TRUE );
  if ( (int) $remove !== 1 || current_user_can( 'edit_users' ) ) return;
  global $wp_admin_bar;
  $account = (array) $wp_admin_bar->get_node('my-account');
  $info = (array) $wp_admin_bar->get_node('user-info');
  $logout = (array) $wp_admin_bar->get_node('logout');
  $account['href'] = $info['href'] = '#';
  $wp_admin_bar->remove_node('my-account');
  $wp_admin_bar->remove_node('user-info');
  $wp_admin_bar->remove_node('edit-profile');
  $wp_admin_bar->remove_node('logout');
  $wp_admin_bar->add_node($account);
  $wp_admin_bar->add_node($info);
  $wp_admin_bar->add_node($logout);
}

Step 3

Using all the code above, an user do not see a link to profile anymore, however, just typing http://www.example.com/wp-admin/profile.php and profile page is shown.
This can be easily solved:

add_action( 'load-profile.php', 'rpa_profile_banned_check' );
add_action( 'load-index.php', 'rpa_profile_banned_msg' );
add_action( 'all_admin_notices', 'rpa_profile_banned_msg' );

function rpa_profile_banned_check() {
  $remove = get_user_meta( get_current_user_id(), '_profile_banned', TRUE );
  if ( (int) $remove === 1 && ! current_user_can( 'edit_users' ) ) {
    wp_redirect( add_query_arg( array( 'pbanned' => 1), admin_url('index.php') ) );
    exit();
  }
}

function rpa_profile_banned_msg() {
  if ( current_user_can( 'edit_users' ) ) return;
  static $show = false;
  if ( current_filter() === 'load-index.php' ) {
    $msg = (int) filter_input( INPUT_GET, 'pbanned', FILTER_SANITIZE_NUMBER_INT);
    $banned = (int) get_user_meta( get_current_user_id(), '_profile_banned', TRUE );
    $show = ( $msg === $banned && $banned === 1 );    
  } elseif ( current_filter() === 'all_admin_notices' && $show ) {
    echo '<div class="error"><p>Sorry, you are not allowed to edit your profile.</p></div>';
  }
}

The previous code check if the user is banned when the profile page is loaded, and if so redirect the user to dashboard adding a variable in the url that allow to show a message that inform user that he/she is not allowed to see profile.

This is how the message appear:

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All code, as a plugin, available in a Gist here.