I've known DBAs to get away with little or no programming skill, but every DBA I've ever considered to be any good had reasonable programming skills at least. One or two I can think of had substantial development backgrounds and were fairly good developers in their own right. There's a fair amount of open-source tooling written by people who work as DBAs in their day job and IIRC the guy who wrote TOAD used to work as a DBA.
Depending on the role you might find yourself writing or tuning queries, writing scripts to automate tasks or consulting on application design. In some cases you may just be minding a bunch of servers through OEM or some other monitoring tool.
Modern 'enterprise' development environments such as .Net or Java are complex enough that a developer can make a career just out of specialising in them. As a DBA, particularly in the development space, having a working knowledge of C# or Java might not hurt, but you probably won't spend a lot of time actually coding in them.
You will probably get more mileage from whatever scripting tools are used on your platform, although a lot of systems expose .Net, Java, COM or web service APIs. If you need to code something against these APIs you will need at least a basic working knowledge of something that can consume that API. However, advanced application architecture skills are usually not necessary to do this.
Some developers will have strong database skills, but irrational fear of databases is quite common in development circles. Many developers also never really get their head around the 'set operations' paradigm that underlies SQL. As a Dev DBA you can find yourself dealing with the consequences of this, and maybe having to intervene in stored procedure code to sort out performance issues.
ETL and tooling surrounding the database may also fall into the remit of the DBA. I've seen quite a few DBA roles advertised that seemed to involve a significant amount of back-end development work. This will be most common in smaller companies. One recent poster wanted to integrate custom metrics into Oracle Enterprise Manager, which has a plugin API to do this. It is quite common to see requirements like this turn up, and essentially the only way to this is to write some glue code.
There are plenty of 'Tools Guys' working in I.T. and they can get useful work done in spite of the parochalism. However, when the tools run out of steam, often the only way to get something done is to actually write a bit of code to do it. This is where programming skills separate the men from the boys.