Monitors doesn't always really contribute to the heat output of the GPU. Unless you have a high resolution monitor in which case you don't.
Checking on the monitor that you have, it only has a resolution of 1440x900 which isn't that much to stress your GPU up.
But it is normal that your GPU would "warm up" while playing more demanding games especially if you have cranked the settings all up. It shouldn't damage the GPU as long as it stays under it's thermal limit. Just make sure you have good airflow inside your case and you should be fine.