In order for the Jaspersoft to start automatically after a restart of the machine, the "service.sh" script must be executed in the Jaspersoft folder. However, only Postgres will be started after a reboot and Tomcat will not.
Auto Start/Stop with Bundled Tomcat and PostgreSQL.
To have JasperReports Server automatically start when you reboot your Linux server, you need to install the JasperReports Server database and application server as services. If you have installed JasperReports Server using the binary installer with the bundled Tomcat and bundled PostgreSQL options, an example jasperserver service script can be found in the following location:
Edit this script and set permissions as described in the /scripts/linux/readme file in the same location. Once installed, these services are started automatically when you reboot. Consequently, the JasperReports Server also automatically restarts.
This is further documented on the JasperSoft Community Documentation page: