Will the computer cause permanent damage
to my eyes?

There have been concerns of possible UV damage from the screen.
The scientific bodies state that there is no evidence of this.

Dry, gritty, sore eyes?

What generally happens is that when you are concentrating on the screen, you tend to blink less. This, in addition to being in a dry air-conditioned office or centrally heated room, can disrupt the tear film, resulting in the above symptoms.