While it's pretty unlikely that a DVD reader could write over a DVD, it's very possible for the mechanicals inside the DVD reader to damage a DVD. Does this DVD reader use a slot loading mechanism? If so I'd say that's probably the culprit - all it takes is a piece of hard grit on the loading mechanism to scratch the living hell out of the media.
Hold the problem DVD up to a bright light, and look through it from the unlabeled side. This makes it very easy to spot scratches in the media (label side).
Edit: wait, just read your post where you say "this keeps happening to this one DVD" - so this is a DVD-RW? If so maybe try rewriting it in a different writer, or at least at a different speed. Or better yet, toss it and use a different disc altogether and see how that goes.
