You can play at 1080p on a 1440p monitor but that won't change the monitor's refresh rate to 144. It will still be a 60Hz monitor.
The other problem is that monitors don't always have the best internal scaling. A 1440p monitor will still have a resolution of 2560 x 1440 when you are playing with the game settings at 1080p. The monitor has to scale the 1920 x 1080 image to fit its 2560 x 1440 screen.
Some monitors are good at this but it most cases you would be better off just playing on a 1080p screen if you are going to game in 1080p. The monitor's scaling process can degrade image quality.