What if you concatenate the string, like
$author_ig = '[instagram-feed id="' . get_the_author_meta('instagram') . '" imageres="full" showheader="true" showbutton="true" buttoncolor="#fff" buttontextcolor="#000" showfollow="false"]';
echo do_shortcode($author_ig);
I used get_the_author_meta instead of the_author_meta, because the former returns the meta as a string while the latter echo’s the meta.