What happens to photons when they hit our eye?
Some of them pass through the iris and are focussed by the lens onto the retina where they are absorbed by rods or cones.
where do they end up?
Some of them end up absorbed by rods/cones, some by other tissues, some are reflected (c.f. red-eye in photography).
why our eye don't get overheated?
It does if you stare at a bright source of light (sun, lasers, arc-welders etc). This causes permanent damage and irrevocable loss of sight (in at least part of the visual field)
The eye includes a mechanism for preventing too much light entering it. This is the iris. When this is overloaded, a secondary mechanism is activated, this is the eyelid.
Temperature in the body is regulated. The mechanisms have their limits. I believe blood flow is one way heat is transferred from place to place in the body.
Similarly what happens to electrons when the light hits certain metals
They may get excited, temporarily. You should probably ask this as a separate question but the answer should be easily found by searching.