List tags by certain letter issue with special chars like å, ä and ö

Actually the code is working, its just hiding the empty tags. So you better populate the tags named with å, ä and ö or you can add the line hide_empty => false to the argument. And then the code will be like below-

<ul>
    <li>
        <h3>Ö</h3>
    </li>
    <?php
    $tags = get_tags(
        array(
            'name__like'    => "ö",
            'order'         => 'ASC',
            'hide_empty'    => false // It will show the empty tags.
        )
    );
    foreach ( (array) $tags as $tag ) {
        if( strtolower( mb_substr( $tag->name, 0, 1 ) ) != 'ö' ){ continue; } ?>
        <li>
            <a href="https://wordpress.stackexchange.com/questions/271316/<?php echo get_tag_link( $tag->term_id ) ?>">
                <?php echo $tag->name ?>
            </a>
        </li>
    <?php } ?>
</ul>

This above code is tested.

Hope that helps.