What bugs would happen because it's running at a higher refresh rate? That usually improves things like collision accuracy, whereas lower framerates make it less accurate.
I think generally if someone has gone and deliberately bought a 120 Hz monitor, they are probably technical enough to know they should have a powerful CPU to match the higher system requirements to reach 120 FPS rendering. They probably also want to have everything render at 120 FPS (otherwise why did they splash out all that money on a specialist display?). I think it would be very disappointing to override their decision and make the extra hardware useless and force them back down to 60 FPS!