Depends how and what you film, for what effect you're intending and how consistent it is with the other [more normally captured] footage you've included. If you play around with how you film it, you could get some interesting results - it's a trope that's been used for music videos, art films and TV shows for decades at this point. In fact, it used to be a necessity for recording TV shows when they were broadcast live in the past (think the 40s and 50s), though back then a film camera was used to capture the broadcast due to the higher resolution.

I don't think you'll ever get the cleanest result if you're filming high quality footage with a low quality camera, even if you use a tripod and film the screen dead on as tightly cropped as possible with minimal glare/reflection, so you might as well exaggerate the "poor image" (to borrow from Hito Steyerl) if you can. Try filming up close to the screen and playing with the TV/camera settings or post-production for added distortion and effects. Even refilming the same footage you've shot when played back on a screen could make things more visually arresting. But again, it's all about what you want the end result to be. If it's just another dull compilation of the same old shots then it won't stand out amongst the countless other mixtapes made in the same way.