There's generally a bit of risk associated with being an early adopter. While it's neat to be among the first to try out the latest technology, first generation products are often riddled with bugs and are the first in line to reach end-of-life status.
That's exactly what's happening now to a long list of devices that have, up to this point, run YouTube's app without incident. Due to changes in the app's API, the YouTube app will (as of writing) no longer work on select devices manufactured in 2012 and earlier.
On the iPhone, iPod touch or iPad, you'll need at least iOS 7 or newer to access YouTube. Second generation or older Apple TVs need not apply as there's no way to continue watching YouTube on these set-top boxes.
Likewise, devices running Google TV version 2 or earlier are out of luck. A selection of older Sony TVs and Blu-ray players as well as Panasonic TVs and Blu-ray players are also being phased out. YouTube notes that if your smart TV, smartphone, tablet or game console supports flash and / or HTML5, you may still be able to continue watching by visiting the main site in your browser.
You'll know right away whether your device is impacted by the API change as you'll see the notice above when launching the app. Your best bet is to either try watching in your browser, find a newer device or ditch YouTube completely.