Do I really need triggers for relational database, for example, PostgreSQL?

There are few use cases where triggers comes handy in relational database. Some of them include audit trail capture, archiving, history/snapshot capture etc.

Regarding whether you would need it or not depends on application you use, the use cases for your business requirement and how well they can be handled by application.

If any of the above stated use cases are involved and application can gracefully handle them without much loss in performance/user experience then it makes sense to implement it in application logic. Otherwise, trigger logic would be better to implement