The Natural Resources Defense Council commissioned a study in late 2015 to understand the energy consumption from 4K TVs, titled The Big Picture: Ultra High-Definition Televisions Could Add $1 Billion to Viewers' Annual Electric Bills. While this doesn't exactly answer your question as regards the resolution of You-Tube videos, it gives some solid clues.
First of all, the study finds that "UHD televisions use an average of 30 percent more power than HD televisions of a similar size." This finding is illustrated in Figure 9 from the report:
So this means that if you have an HD or UHD capable screen, you are likely using more power than a screen which is not capable of showing higher resolution video.
Next, the authors compared energy consumption on the same TVs for HD vs 4K content: "Feeding a 4K television, a higher-resolution (4K) version of HD video material did tend to increase energy use by an average of 10 percent, but there were large differences between models." Figure 16 illustrates this:
Thus, it seems that when you are viewing higher resolution video you will indeed be consuming more energy, but the exact difference varies significantly by model.
Finally, the authors find a bit of a silver lining: "Meanwhile, streaming higher-resolution content via the Internet instead of playing it from a disc did not make a significant difference in energy use, which is good news as people increasingly stream content over the Internet."