Going from 144 to 240?
There was another thread where there was a long-running debate about that. I'll say here what I've said there: Until they can prove, in a repeatable, scientific process, double-blind study, that people can tell what refresh rate they're playing on, and reliably prove better play when they don't know what the refresh rate is, I'll believe it.
Until then, given that the record fastest human response to anything was 13ms, well...
1 sec / 13ms = 76.923 frames/sec. I'd go so far as to accept maybe 80-100 is where the ability to perceive and react tops out, maybe.
Ergo, my conclusion that going for faster than 144Hz doesn't help. Faster than 100, even, I'd say, but since 144Hz monitors are around the same price as there slower brethren (given otherwise identical specs), there's no harm in getting a faster monitor at the same price.