Try a glob of Vaseline on your temples. Some runners say it works or diminishes the problem and there is no harm if it doesn't. I've not tried it.

I find the problem worse at the end of a long hot ride when I don't drink enough and believe the intensity of the burn is proportional to the salt concentration in the sweat. If I hydrate well before and during a ride it burns less or I don't notice it as much.

Dry skin is one of my problems too and sweat can burn when the skin around my eyes is dry and/or irritated. Wiping my eyes often doesn't help. Regular use of an industrial strength moisturizer like CeraVe helps.

Anything to stay cool - pouring water down my back or over my head, soaking a skull cap in cold water, etc. Filling bottles with ice at a convenience store is wonderful. Some guys use ice socks.

https://en.wikipedia.org/wiki/Ice_sock


-Tim-