Yes and no. Your GPU will still output more frames than the monitor's refresh rate can display. However you won't be able to see all of them, thus diminishing the value of your expensive GPU. So if you are capable of running a game at 120 FPS but your monitor is only 60Hz you will only see half the frames you are drawing.
You can also run into issues with screen tearing, lets say you are generating 70 FPS on a 60Hz monitor, some frames will draw halfway on the screen before it refreshes and begins the next frame, resulting in a torn picture. Enabling Vsync can fix this, but this will cap your FPS to 60 max or 144 max (depending on your display).
TL;DR It's a waste of your money to use a GTX 1070 or 1080 on a 1080p 60Hz display, you might as well get a GTX 1060.