Are there massagers useful in reducing eye fatigue, eye strain, or dryness?

Rest, lubricate. Resting your eyes between pages on book or computer is good idea. Eyes blink less when reading. The blinking allows for upper lid to grab some tears resting on lower lid. Add artificial tears preservative free drops.
No. In fact, massaging or rubbing the eyes can be detrimental in the long run. Patients have developed scars and other injuries from eye rubbing, and there is some evidence that children who rub their eyes constantly may have an increased risk for nearsightedness and keratoconus. Your best bet is see your eye doctor in order to figure out the cause of the strain or fatigue.
No. Our eyes are designed to see things in 3d. When you look at a computer/book etc for a long time (a 2d surface) you can strain your eyes or get dry eyes. Every 20 minutes rest your eyes for 20 seconds by looking at something at least 20 feet away.
Massage the lymph. Massage the sides of face, mandible, clavicle and scalp in increase lymph. Eye stress is due to liver, bags under eyes is kidney so unless insides reflect health - face rx alone would not help.