Does a DBA need to know how to program in a system language besides SQL? To what extent does a Database Administrator need to know system or application level programming languages (for example .NET or PHP) besides "just SQL"?

For the purposes of this question, no specific version of the SQL standard is considered for this answer (SQL ANSI 86, SQL ISO 87, SQL:2008) as the question is in regards to desktop or server languages outside the realm of SQL.
It depends.

In a big shop, maybe not because you have 1000s of servers to look after and your tools are provided. In a small shop, you'll probably need more knowledge because you have a wider remit.

Scripting languages (PowerShell, cmd.exe for example) are always useful though for monitoring, deploying etc. I've often had some Perl scripts I had to maintain (just about). Then there are the assorted ETL packages that you'd be expected to know about.

Saying that, most DBAs (that I know) would be able to write basic CLR stuff or know PL/SQL well. For me, the dividing line is knowing about the broader patterns or architecture of .net or Java. I don't need it as a DB developer or DBA. In the same way that .net or PHP or Java Monkeys don't understand database design or architecture or code as well as I do.

Personally, I decided to stop chasing the latest or best client language years ago and focus on database work. That doesn't make me a "non-programmer": I'd need to learn it again if I had to.