No, QuickBooks Desktop itself does not need to remain open for scheduled automatic syncs to run. However, the QuickBooks Web Connector does need to be open for it to run the scheduled automatic syncs with Funding Gates. You can schedule your automatic syncs within the Web Connector.
However, if you are manually syncing through the Web Connector, by clicking 'Update Selected' for the FG Receivables Manager, QuickBooks must be open. If QuickBooks isn’t open you’ll likely see the error message 'Application requested to abort'. If this happens, simply open QuickBooks Desktop and run the sync from the Web Connector again.