Backups
From my experience a lot depends on how the database is installed and packaged with the application. If you sell an application to a customer with the database layer implementation details hidden or abstracted, then the typical customer expectation is that a backup solution is provided as part of the software. This expectation is there for a variety of reasons.
Off the top of my head, one of which is that the IT department supporting the application cannot be confident in how the backup of the db should happen for the application. Sometimes, a DB dump with some vendors can cause locking issues that interrupt normal application operation. I'm sure you can imagine other complaints an IT department might have for an application that bundles a DB.
As for installs that connect to a existing DB, I think it is a reasonable expectation that the DB admins for those databases handle backups. However, it is very important that you - the application provider document what needs to be backed up. Are there sequences and indexes that need to have their precise order persevered? Are there terabytes of data created that may not be important to your customer if lost? What about the state of the application during the backup (as mentioned above)? Does the application need to be shutdown?
Restores
More problematic is restores. What if a partial restore of data is needed? How could this corrupt your data store? Does your application store data anywhere else (files, network) that can be put in an unreliable/broken state as a result of a historical backup?
Anyways, a long story short - when I'm a customer, I feel a lot better knowing that the application vendor has QAed the backup/restore process and provides support. Actually, the was a requirement of a large customer of my current project. They were more than capable of managing a database, rather they wanted us to take responsibility for our own data and guarantee that the process would be flawless.
In the end, your mileage may vary depending on the implementation, requirements and industry expectations of your specific application.