VisIt doesn't really matter, I've always played with 60hz and everyone who has upgraded to 144hz from 60 said it's different but you don't just straight out improve. Also your eyes will eventually get used to the new refresh rate and if you had a better performance at the beginning, it will slowly decline, so I assume the same would apply from 144hz to 240hz. Unless there is some game out there that requires it, I wouldn't recommend upgrading.
its not just about "smoother image" or your 'eyes getting used to it' its about the input lag/frame delay as well. on 60 hz there's 16.67 ms between each frame, 120 hz that halves to 8.333, 240 that's down to 4.16 ms