Is there a way to export to a CSV file but without the header row containing the field names? (It isn't a huge deal to edit the file and remove the header row, but it would be nice if we didn't have to.)

No, this is not possible. You can however copy & paste the selected cells without a header.

Any update on if this is a feature in Pl/SQL Developer v14 or greater or if there is a suitable plugin? If not, can we add it as an enhancement.

Working with Oracle Fusion Enterprise Resource Planning (ERP) and one of the import files requires no header.

Manually removing it is possible but trying to get the export format correct with minimal to no changes required.

There is no Plug-In for this. I have added this to the list of enhancement requests.

The "User-defined Text Export" plug-in can do this. With that plug-in, you will have a new "User-defined file" export method with an option to include headers.