Since our eyes evolved in a constant 1G environment, and the fact that the vitreous humour only weighs a few grams, the orientation of our eyes while doing any activities should not be a factor what so ever. If anything, lying on your back could possibly aid in that the mass of the vitreous humour is more evenly spaced out on the photo-receptors as opposed to graduated when standing/sitting upright.
Of course, once we get outside the constant 1G, walking/running speed environment, all bets are off.
As all the related questions have shown (Does watching television damage the eyes?Does watching television damage the eyes? and http://skeptics.stackexchange.com/questions/1092/is-it-harmful-to-your-eyes-to-read-in-dim-lightIs it harmful to your eyes to read in dim light?), the eye is a versatile and resilient organ. Most of these questions are the result of misrepresentations of other physiological effects, that have grown into "old-wives-tales".