Unless you have v-sync enabled, which will limit your maximum frame rate to your monitor's refresh rate, your system should push the same frame rates no matter what refresh rate your monitor has.
Of course, if you have a 60Hz monitor, it will only update the image 60 times per second. Your system may be rendering more frames that that, but the monitor will only display the most recent frame, and any extras will just not get used. A higher refresh rate monitor can display those additional frames that would have otherwise been "thrown away", resulting in smoother motion, provided your system is actually pushing more than 60fps.
And likewise, a 240 Hz monitor can display even more frames if your system is pushing more than 144fps, though you tend to run into diminishing returns as far as how much smoother a game will appear, and most recent games won't be pushing frame rates that high.