Hello Eyera
Basically, the CPU socket is correct, but the chipset is the problem. Intel 8th Gen CPUs require 300 chipsets (
https://www.intel.com/content...). Ergo, you need a new mainboard.
The question if at all: Basically, there are two types of bottlenecks in gaming: Either GPU or CPU.
High resolution and high details require more graphics power, so the GPU is usually the bottleneck - the CPU can easily keep up with the (relatively low) frame rate.
Low resolution (as in your case with 1080p) and low (depending on the game also high) detail settings mean less work for the graphics card, which results in a high frame rate. At some point, however, the CPU will have trouble keeping up.
Of course, this all depends on the game. But as you can see, if you want to achieve high FPS (which is what you want with your 240Hz monitor), the CPU is more likely to become the bottleneck.
In summary: Yes, it would be worth it.
Then there's the question of which CPU. With the H170 chipset board, the max new CPU would probably be the i7 7700K (double check). If you want newer, you'd have to get a new motherboard - then you could consider waiting for the new Ryzen (should be coming soon), which would probably be the most interesting in terms of performance.