Having a lower fps than the hz of your monitor is fine, and will not cause any issues. The 144 Hz is maximum that the monitor will handle, so playing 50 fps in Crysis 3, just means that you will be hitting 50 fps while playing. Getting a monitor that is 144 hz is only worth doing if you plan on using applications or games that you will be reaching that high of an fps.
When you are talking about getting input lag unless you are getting 144 fps in each game, that is a tricky question. If you are playing Crysis 3 and getting 50fps in it. Then you can still end up getting fps lags/spikes that cause your fps to drop lower...or even get fps that hits higher. If you have a 60 hz monitor and have vertical sync enabled, then the system will try and maintain as close to your monitor refresh rate in fps as possible...which would help maintain that 50fps as best as it could (but you could still get lag spikes). But if you were getting fluctuations of 50-70 fps with a 60 hz monitor, that could cause screen tearing, so the system would try and cap you off at 60 fps, and maintain that as well as possible.
By choosing a monitor that is 144 hz, that same principle will be used. So for some games that you may turn down the settings for, you will reach higher fps, and can hit 144 fps or greater, so v sync with that monitor would prove useful and give a more smooth and better experience. However even if you decide to go with a monitor that has a higher refresh rate/hz than your games would be able to play, you could always use either the in game or other program settings to set a fps limit for the game. That way you can minimize screen tearing issues and lag spikes.
So, V sync only really makes sure the system is not running the game or program at a higher fps than your monitor can handle. Which will help your system run cooler and better, since if it could push games at 70 fps, but your monitor will v sync cap it at 60 fps, then the GPU and CPU will not continue to push 70 fps to the monitor. In addition to better performance and cooling, you will also limit screen tearing.