Will he be sinning if he continues to use the computer although it is damaging his sight?

Praise be to Allah.

It is not permissible for you to continue to neglect your eyesight and cause damage to it. Even if you do not care about something that will cause harm to you, Islam does not approve of it. The Prophet (peace and blessings of Allaah be upon him) forbade all kinds of damage and harm, as he said: “There should be neither harming nor reciprocating harm.” Narrated by Ibn Majaah (2340). 

Al-Shaatibi (may Allaah have mercy on him) said in al-Muwaafaqaat (3/16): 

Harming and reciprocating harm are forbidden in sharee’ah, and there is a great deal of evidence to that effect, such as n the verses (interpretation of the meaning):

“But do not take them back to hurt them”

[al-Baqarah 2:231]

“and do not harm them”

[al-Talaaq 65:6]. 

There is also the prohibition on transgressing against people, property and honour, and robbing and wronging others. All of that is included in the meanings of harming and reciprocating harm. That also includes offences against people, or causing harm to their mental health, offspring or property.  

So it is a concept which is very broad in application in sharee’ah. End quote.  

Shaykh Ibn Baaz (may Allaah have mercy on him) said: Everything that harms a person with regard to his religious commitment or worldly concerns, it is haraam for him to consume it, whether it is poison, tobacco or anything else that is harmful, because Allaah says (interpretation of the meaning):

“and do not throw yourselves into destruction”

[al-Baqarah 2:195].

And the Prophet (peace and blessings of Allaah be upon him) said: “There should be neither harming nor reciprocating harm.” End quote.

Majmoo’ al-Fataawa (6/23-24). 

Strive to follow the doctors’ advice in the way you use the computer, and pay attention to the means that will not harm your eyes. Seek forgiveness from Allaah for your past heedlessness, and do not neglect your health or neglect yourself in the future. All of this is by the blessing of Allaah which you should preserve. Then if it continues to harm your eyes, after you have taken all protective measures, then there will be no sin on you, in sha Allaah, rather you will be rewarded if you are patient and seek reward with Allaah. 

We ask Allaah to grant us and you health and well being. 

See also the answer to question no. 107305. 

And Allaah knows best.