Which areas of a DBA's knowledge should a developer delve into?
The two things I deal with every day.


Disaster recovery.
Performance tuning. (Both for individual queries, and for the dbms itself.)


Your disaster recovery plan needs to be 


scripted, 
tested, and 
practiced.


I'm using scripted in the sense of something an actor would follow, not something written in Python. It should tell everyone who needs to be involved exactly what to do. (And often, exactly what to say, too.)

Performance tuning for queries includes understanding keys, indexes, and normalization. (Often "tuning" problems are actually structural problems.)