Delete option value from array using update_option()

There are some problems with your code, below code should answer your question how to delete, remove the subscribers from the options array. I heavily commented the code, which should explain what I did:

Add form:

//add the forum subscription option as part of the personal options on the profile and user edit screen
add_action( 'personal_options', 'forum_subscr_to_profile_form' );
function forum_subscr_to_profile_form($user_obj) {
    //get user ID from user object
    $user_id = $user_obj->ID;
    //get option into variable
    $forum_subscr = get_option('forum_subscr');
    //check if user is in subscription array
    $fs_is = isset($forum_subscr[$user_id]);
    //depending on that setup variable for input value 
    $chkb_val = ( $fs_is == 1 ? 'false' : 'true' );
    //depending on that setup variable for checked staus
    $chkb_chk = ( $fs_is == 1 ? 'checked="checked"' : '' );
    ?>
        <!-- table row for the additional (small) header -->
        <tr class="additional-user-profile-opt">
            <th scope="row">
                <strong><?php _e('Additional Options'); ?></strong>
            </th>
            <td></td>
        </tr>
        <!-- table row for the option: title - checkbox - description -->
        <tr class="forum-subscribe">
            <th scope="row">
                <?php _e('Forum Subscription')?>
            </th>
            <td>
                <fieldset>
                    <legend class="screen-reader-text">
                        <span><?php _e('Forum Subscribe') ?></span>
                    </legend>
                    <label for="subscr_chkb">
                        <!-- choose a non-generic name to prevent conflicts, namespacing/prefixing it would be even better -->
                        <input id="subscr_chkb" type="checkbox" name="forum_subscrbd" value="<?php echo $chkb_val ?>" <?php echo $chkb_chk ?> />
                        <?php _e('Subscribe to Forum') ?>
                    </label><br />
                </fieldset>
            </td>
        </tr>
    <?php
}

Add/remove data:

//take care of adding and removing the user subscription from the array 
add_action( 'personal_options_update', 'forum_subscr_add_remove_user' );
add_action( 'edit_user_profile_update', 'forum_subscr_add_remove_user' );
function forum_subscr_add_remove_user($user_id) {
    //get user object
    $user_obj = get_userdata($user_id);
    //get option into variable
    $forum_subscr = get_option('forum_subscr');
    if ( !isset($_POST["forum_subscrbd"]) || $_POST["forum_subscrbd"] == false ) {
        //remove element from array by
        //first setting the value to null
        $forum_subscr[$user_id] = null;
        //and then filtering with array_filter
        $forum_subscr = array_filter($forum_subscr);
        update_option( 'forum_subscr', $forum_subscr );
    } else {
        //add element to array $user_id as key and $user_obj->user_email as value
        $forum_subscr[$user_id] = $user_obj->user_email;
        update_option( 'forum_subscr', $forum_subscr );
    }
}

Keep data up to date:

//make sure the email in the subscription array gets updated when the email gets changed
add_action( 'show_user_profile', 'forum_subscr_keep_email_uptodate' );
add_action( 'edit_user_profile', 'forum_subscr_keep_email_uptodate' );
function forum_subscr_keep_email_uptodate($user_obj) {
    $user_obj = get_userdata($user_obj->ID);
    $forum_subscr = get_option('forum_subscr');
    //update the email for subscribers if the profile page and the subscription array email are not equal
    if ( isset($forum_subscr[$user_obj->ID]) && $user_obj->user_email != $forum_subscr[$user_obj->ID] ) {
        $forum_subscr[$user_obj->ID] = $user_obj->user_email;
        update_option( 'forum_subscr', $forum_subscr );
    }
}