Restoring PostGres is not sufficient for restoring a Tableau Server environment. You will need to run Tableau Server backups in order to be able to recover a working Tableau Server instance. This can be done using the "--unsafe" flag if you do not want to stop the server for the backup process (see the backup guide in the online help). These backups should contain all you data with the exception of workbooks or extracts which are at that time, in the middle of being published updated or refreshed. Usually, in the event of a disaster these can simply be re-published individually.
Additionally, Tableau Server now has "High Availability" which means that a single environment can have passive, fail-over PostGres and Data Engine components (see "Understanding High Availability" in our online help).