No. There is a condition termed "computer vision syndrome" which describes eye strain and fatigue after staring at computers, and can be easily treated with eye drops and (occasionally) glasses. Staring at any object for a long time causes the same temporary condition, but we spend so much time using computers that we tend to think they are leading to vision loss rather than strain from overuse.
No. Although using your eyes does not damage them from wear and tear, you can have discomfort and blurred vision (eye strain) if you need glasses or if they need updating or if you have a propensity for dry eye.