If I use this shortcode [acf field="{$field_name}"] the field doesn’t render. What’s wrong?
There are a couple of things I can think of without having more information.
1) You’re putting the shortcode into a field that does not support them
2) You’re using the sortcode in a different place than were you putting them, in other words you’re trying to add a custom field that is associated with a different post, an options page, a user or a term. In this case you’ll need to supply the correct post ID value.