I don’t see anything in your code that would keep it from working. Not sure what’s going on. I would try using === instead of ==
<? if( get_field('post_label_status') === 'Sponsored' ){ ?>
<div class="post-status-label-feed">
<span>Sponsored Content</span>
</div>
<? } ?>
<? if( get_field('post_label_status') === 'Opinion' ){ ?>
<div class="post-status-label-feed">
<span>Opinion Piece</span>
</div>
<? } ?>