Hey Zay,
It depends on if you want to run the games at 240 Frames per second. That would be very difficult even with a top end graphics card. Otherwise, your 240HZ refresh rate is (kind of) going to waste. To even get close you would probably need a top of the line CPU as well to prevent one maxing out before the other (however that depends greatly on the demands of the game). If you truly want 240 FPS then I would probably recommend a 2080TI and that would only get you close to around 200 on most modern games at 1080p. To get 240, you may need to SLI them, or hope for a beastly overclock to your card.
To be honest, 144HZ @144fps is pretty standard for pros and hardcore gamers. The benefits of getting up to 240FPS/HZ are negligible. I hope that helped, and wish you all the best in your build!