Yes and no. Usually not on youtube (though is supposedly possible by using an H.264 browser plugin or is that native now in Firefox??), but using other video playback apps that can use DXVA or CUDA acceleration, the decoding can be much faster (though only needs to be fast enough for realtime playback). This can help reduce tearing artifacts when upsampling to a larger window on a high resolution (like 4K), and offloads more processing from the CPU. It also allows more decoding quality than software-on-CPU does because a CPU is inherently limited in performance at something like this.
Additionally when upsampling video if you have the extra GPU horsepower you can apply shader filters in real-time to improve image quality such as denoising, edge sharpening, or more complex sharpening algorithms.
The break point for making this worthwhile might be upsampling 720p to a 24" or larger 1080p screen, or anything 2K or lower up to a 4K screen. In other words if you don't have a nice high pixel screen, you should upgrade that first, keeping in mind that his is another benefit of a newer, more powerful card, that some can now do 4K at 60Hz by having HDMI2 output. With only HDMI1, let alone DVI, you are stuck at 30Hz at 4K resolution which is too low for comfort.
I'm suggesting that if your monitor is as old as your video card, the monitor might be the place to start first and if you make it 4K, then you "need" a new video card for 60Hz refresh rate at everything, desktop use too, not just youtube videos or gaming.
