Someone asked this on Reddit, to no conclusion.
In the source novel his logs are in text format, not video. At one point (after retrieving Pathfinder) he makes an assertion that he should be more careful with his logs,
now that someone might actually read them
However, ultimate fate of his logs is left undisclosed in both media.
And when you think about it, his logs only hold sentimental significance. Was he dead, NASA would try to rescue his logs (with his body) to collect information. But since they have a living Mark Watney, they could just ask whatever it is they're curios about (Potatoes? On Mars?!) directly to him.