To start - I'm not an economist, however I am a software developer and have grown up with a technology and education background.
In the short term (5-10 years) I don't see this being that big of a threat. We may be capable of it soon but actual implementation takes a really long time, especially on a large enough scale to hit the entire economy. Hell, a lot of businesses are still using software that requires Java 6, or a really old version of IE. That doesn't even count how much software hasn't been updated from XP but just "hacked" to work with Windows 7. With that said, the jobs will not go away, just shift like they always have. The shift may be more substantial this time, but it's still a shift.
For example, Taxis.. I don't see people fully trusting taxi's to drive us around for at least another 8-15 years. The Taxi driver will just need to be educated on how to override it when/if it does mess up and they could slowly shift toward more of a personal assistant role or entertainment for the drive. I know it's almost unheard of now-a-days, but offering a certain level of (human) customer service will go a long way. For the economic side - people that repaired VHS and other outdated tech also had to adapt. You can't blame DVD or Blueray for it, it's adapt or die - aka life.
Personally I feel like the problem isn't the AI and robotics that are on the way, it's the crap that we call an education system. High school is a complete waste of time, literally a repeat of middle school with added bull@#$% complexities that don't actually help outside of high school. Teachers are often simply baby sitters and aren't allowed to teach outside of the pre-defined curriculum. On the other side, college is ridiculously expensive and doesn't guarantee a job when you graduate, nor a legitimate education depending on where you go. Hell, the closest university to me teaches intro to Java/Programming on PAPER, as in you have to write out your program with a pencil. All of this technology and change requires an education to keep up, but the baseline for education seems to be dropping. Sure, there is Youtube which has been more of a game changer than most people probably know. People spend a few minutes to an hour of their time to upload a video and suddenly millions can find it and learn from it. Want to learn C++? Youtube it. Want to learn how to bake a cake? Youtube it. I bet you can learn more in an hour online than 2-3 weeks in a college classroom.
TL;DR: We aren't doomed like they always want us to believe. Some things will have to change though or it will only get worse. It's going to actually take some teamwork for once and no more of this "I hate you because you're [Insert political party/religion/sexual orientation/etc. here]".
Side note: I would recommend picking up any kind of basic scripting language at the least, regardless of your current profession, consider it an insurance for the future and a tool to increase your own productivity for the present. Applies less to physical labor jobs, but still a good skill to have regardless.