We can see this as an increased demand for education / training. But, not everyone is able to keep up and over time ever fewer people are going to be capable of the remaining niches. It's possible that the service industry's are going to continue to absorb the less capable, but that does not help the economy in the long term.
One of them has his or her hand in your mouth and might need to drill, remove, fill or otherwise mess around with your teeth. I think a lot of people would be more comfortable with a human element in all medical interactions, even if robots are eventually as effective.
What about when robots become more effective? Drastically more effective?
I would not let a modern day robot work on my teeth, they are too stupid, but I can imagine a day (fairly soon) when this will not be the case, and my gut reaction will start to be the exact opposite: I would not let a human being work on my teeth.
Now, with automation and 3D imaging technology we can have a specialist specify what to change and let a computer design a series of discrete non-invasive attachments that allow a home users to quickly attach and remove their implant. It's far less painful, takes less time,can far more precisely preform complex work like rotating a tooth, and the only downside is it costs about the same amount as braces. Note: This is an actual company not just BS (http://www.invisalign.com/Pages/default.aspx).
PS: I still occasionally see a teller for complex interactions, but I trust an ATM to be far more accurate for my day to day needs. And there is still plenty of work for orthodontists, but a lot of the simple stuff is simply better handled by a machine.
As to programming; I don't know a lot about how the software works, but advanced users in front of really complex software like Excel and Photoshop tend to blur the line between a Specialist and Programmer.
Yes, I would save money by going to the robot dentist, but my dentist would be out of work in no time. It partially rests on humans to keep the exploits of technology at bay. We can't turn personable human tasks into machine work.
Our world can be set up to be completely self-sufficient. Yes, a few people would make a lot of money, but I personally believe the impacts would be horrendous.