Let's say I am playing a game with my GTX 960 and i5-3570k @4.5 GHz. There are other things going on in the background like rendering a video, streaming, etc, any cpu tasks. Will a 970 helps increase my fps in this situation?
eh... it depends. Like for video rendering yes a power powerful card will cut some time off. streaming won't make a difference even at 4k and most cpu tasks will have zero impact when comparing a low end card to a high end. As for games, yes you will see a benefit of using a more powerful card, thus more FPS.
If the video renderer is using the gpu then no matter what gpu you have, your fps will drop because it will hog all the resources it can get. If it's cpu tasks and your fps drops, then it's a cpu issue and gpu won't matter. Gaming is one of those tasks that uses everything and if one thing is being used you're going to drop fps and there is no piggybacking on another piece of hardware. Everything has a different job hence why different components exist.