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1) If you've created a dashboard using postgres connection and publish it to server, the data refreshes given that Tableau Server has access to Postgres DB. Doesn't matter if you've shutdown your desktop, once you publish the workbook to server the connection now exists between Tableau Server and underlying data source of your workbook. As long as those two are connected you need not worry.
2) If your data source is an excel file, it is best practice to keep this file in a network location where the Tableau Server has access to. Also, make sure you use UNC path when connecting to excel file through desktop and uncheck "Include External Files" when publishing workbook to Tableau Server.
Here are some links,
Hope this helps. Let me know if you have questions.