When placed in the theme functions.php file this will add a checkbox to the 'Form Options' part of the Form Settings page which when checked will cause entries to be automatically deleted at the end of the submission process.

RW_Delete_Entry.php

<?php
class RW_Delete_Entry {

    function __construct() {
        if( ! property_exists( 'GFCommon', 'version' ) || ! version_compare( GFCommon::$version, '1.8.5.8', '>=' ) )
            return;
        add_filter( 'gform_tooltips', array( $this, 'add_delete_tooltip') );
        add_filter( 'gform_form_settings', array( $this, 'add_delete_setting' ), 10, 2 );
        add_action( 'gform_pre_form_settings_save', array( $this, 'save_delete_setting' ), 10 );
        add_action( 'gform_after_submission', array( $this, 'maybe_delete_form_entry' ), 15, 2 );
        add_action( 'gform_user_registered', array( $this, 'delete_form_entry_after_activation' ), 15, 3 );
        add_action( 'gform_user_updated', array( $this, 'delete_form_entry_after_update' ), 15, 3 );
    }

    function add_delete_tooltip( $tooltips ) {
        $tooltips['delete_entry'] = "<h6>" . __( "Enable Entry Deletion", "gravityforms" ) . "</h6>" . __( "When enabled, the entry and any uploaded files will be deleted at the end of the submission process. If the form has a User Registration feed the entry will be deleted once the user has been activated/updated.", "gravityforms" );
        return $tooltips;
    }

    function add_delete_setting( $settings, $form ) {
        $enable_entry_deletion = ( rgar( $form, 'deleteEntry' ) ) ? 'checked="checked"' : "";
        $settings['Form Options']['deleteEntry'] = '
            <tr>
                <th>' . __( "Delete entry", "gravityforms" ) . ' ' . gform_tooltip( "delete_entry", "", true ) . '</th>
                <td>
                    <input type="checkbox" id="delete_entry" name="delete_entry" value="1" ' . $enable_entry_deletion . ' />
                    <label for="delete_entry">' . __( "Enable entry deletion", "gravityforms" ) . '</label>
                </td>
            </tr>';
        return $settings;
    }

    function save_delete_setting( $form ) {
        $form['deleteEntry'] = rgpost( 'delete_entry' );
        return $form;
    }

    function maybe_delete_form_entry( $entry, $form ) {
        $feed = function_exists( 'gf_user_registration' ) ? gf_user_registration()->get_single_submission_feed( $entry, $form ) : array();
        if ( empty( $feed ) ) {
            $this->delete_form_entry( $entry );
        }
    }

    function delete_form_entry_after_activation( $user_id, $user_data, $entry ) {
        $this->delete_form_entry( $entry );
    }

    function delete_form_entry_after_update( $user_id, $config, $entry ) {
        $this->delete_form_entry( $entry );
    }

    function delete_form_entry( $entry ) {
        $form = GFAPI::get_form( $entry['form_id'] );
        if ( rgar( $form, 'deleteEntry') ) {
            $delete = GFAPI::delete_entry( $entry['id'] );
            $result = ( $delete ) ? "entry {$entry['id']} successfully deleted." : $delete;
            GFCommon::log_debug( "GFAPI::delete_entry() - form #{$form['id']}: " . print_r( $result, true ) );
        }
    }

}
new RW_Delete_Entry();

I just tried to install and run this as a plug-in but was unable to activate it. I got the following error: Plugin could not be activated because it triggered a fatal error. Parse error: syntax error, unexpected '<' in /home6/colorbo9/public_html/wp-content/plugins/Gravity-Forms-Delete-Entry/Gravity Forms Delete Entry.php on line 12.

Any ideas?? I really need this functionality!!