"He doesn't live away from the beach" would mean that he actually lives ON the beach, and might suggest that he would have some problem with not living on the beach. (For example, "A whale doesn't live away from the ocean.") This would not be a common phrasing. If you mean that he lives on the beach, then you would probably state the positive, "He lives on the beach."
"He doesn't live far away from the beach" suggests that while he may not live on the beach itself, he lives nearby. However, if it is an isolated statement, it might be more common to take out "away" and say, "He doesn't live far from the beach." "Far away" gives emphasis to the distance, and since you're actually saying the opposite, you don't need it. You could also state it in the positive: "He lives near the beach."