i5 3570k to i7 6700k means also new motherboard and new RAM.
For vast majority of games, you won't see big improvements by only changing to i7 6700k while keeping your current GPU.
For very few cpu intensive games, yes, you will see noticeable FPS increase.
Here: less effective improvement for the money.
Going for GTX1080 from R9 290 will have huge improvement on almost all games.
Here: more effective improvement for the money.
More to that, i5 3570k is still quite strong. Only very few today's new AAA games can really be bottlenecked by i5 3570k in a way you can notice it.
I do no think it is already the time yet to replace i5 3570k.