Is it possible to get eye herpes or other sexually transmitted disease from getting infected semen in your eye?

Yes...sort of: Herpes, hpv and syphilis are passed by skin to skin contact not from fluid transmission. Passing these diseases in the way you describe is unlikely. Other diseases like gonorrhea, chlamydia and HIV are passed via fluid transmission from one person to another. Mucous membranes around the eye can become infected. In fact, chlamydia is a common cause of infant blindness acquired during delivery.

Yes: Although to my knowledge that study has never been performed. It would seem likely that exposure of a mucous membrane such as the eye to infected semen could result in transmission of disease.