Proceed with the following steps to add JavaScript to custom field description:

1. Log into Jira with a user that has a Jira administrator's permission.
2. Navigate to Administration > Issues > Custom Fields.
3. Click on the Edit link for the custom field.
4. Enter the following JavaScript snippet into the Description text box, and click on Update. You will need to substitute it in your custom field's ID:

<script> 
 AJS.$('#customfield_10103 option[value="-  1"]').remove(); 
</script>

The following screenshot shows that the Team custom field:

As you can see, the None option is no longer available.