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.