Does increasing "Screen refresh rate" damage monitors?

Low refresh rates are only problematic on CRTs (where they cause flickering), not on LCDs.

Because of this, most LCDs only support a single fixed refresh rates, usually 60 or 75Hz. I don't think you could break one by trying to set it higher; it would simply refuse to work or ignore the setting.

With CRTs, there is a potential for damage if you exceed the maximal refresh rate the hardware supports, but modern CRTs support very high rates (usually at least 100Hz) and probably also have safeguards to protect against misconfiguration.

The concept of refresh rate, is applicable only to CRT, as the refresh rate is an indication of how many times per second the electron beam scans and draws the data. The higher the refresh rate, more number of times the beam is drawn. I can clearly make out the "flicker" at a refresh rate of 60Hz, and it indeed causes severe headache and eyestrain. It's recommended that you set it to as high as your monitor allows, if not atleast 72/75Hz.

As for damaging the monitor, most newer monitors simply will refuse to operate if set at a refresh rate beyond the capability, you will get an error message like Sync out of range.

No risk.
If the monitor supports higher refresh rates then by all means try them. Not everyone thinks higher is better, I'm quite comfortable at 60Hz even though my monitors supports 75Hz.

One thing though. If you have a more than one monitor on your desk make sure they have the same refresh rate. If they don't I can almost guarantee that you'll get a headache.