Is it obligatory to wash inside the eye when doing wudoo’?

It is not prescribed to wash inside the eye when doing wudoo’ or ghusl; rather doing that is harmful to the eyes. 

Imam ash-Shaafa ‘i (may Allah have mercy on him) said – concerning the verse in which Allah, may He be exalted, says (interpretation of the meaning): “When you intend to offer As-Salat (the prayer), wash your faces and your hands (forearms) up to the elbows” [al-Maa’idah 5:6] –: I do not know of any difference of opinion concerning the fact that the face that must be washed when doing wudoo’ is the outward surface of the face, not what is underneath; a man is not obliged to wash his eyes or sprinkle water under his eyelids.

End quote from al-Umm (1/40) 

Ibn Qudaamah (may Allah have mercy on him) said: Some of our companions stated that one of the Sunnahs of wudoo’ is to wash inside the eyes, and that it was narrated from Ibn ‘Umar that he became blind because he often made the water enter his eyes. Al-Qaadi said: Rather that is encouraged in the case of ghusl. This was stated by Ahmad in several places. That is because ghusl following sexual activity involves more extensive washing, as it involves washing the entire body, and washing underneath thick hair (so that the water reaches the skin), that which is beneath the eyelids, and so on, and inside the eyes comes under the heading of parts of the body that it is possible to wash. If it is not obligatory, then at least it is encouraged (mustahabb). 

However the correct view is that this is not Sunnah either in wudoo’ or in ghusl, because the Prophet (blessings and peace of Allah be upon him) did not do this or instruct anyone to do it, and it is harmful. What is mentioned from Ibn ‘Umar indicates that it is nor right, because it caused him to lose his sight, and doing anything that it is feared will cause one to lose one’s sight or will damage it, when there are no reports to indicate that it is prescribed, if it is not haraam then it is at least makrooh.

End quote from al-Mughni (1/77) 

Based on that, it is sufficient to you to wash the face, without getting water into your eyes. 

And Allah knows best.