The batter hit a foul ball that hit his own player? Is the runner out?
I'm going to assume you're asking about MLB Rules.
The main rule covering this is 5.09 (b)(7) and the definition of FOUL BALL is also relevant.
5.09 (b) Any runner is out when ...
(7) He is touched by a fair ball in fair territory before the ball has touched or passed an infielder. The ball is dead and no runner may score, nor runners advance, except runners forced to advance. EXCEPTION: If a runner is touching his base when touched by an Infield Fly, he is not out, although the batter is out.
A FOUL BALL is a batted ball that settles on foul territory between home and first base, or between home and third base, or that bounds past first or third base on or over foul territory, or that first falls on foul territory beyond first or third base, or that, while on or over foul territory, touches the person of an umpire or player, or any object foreign to the natural ground.
A fielder would never be expected to be between third base and the batter, so the situation depends only on where the runner is standing. If the runner is in fair territory, it would be an out. If the runner is not in fair territory, then it is just a foul ball.
Because of this, all runners on third would be expected to make sure they stay outside fair territory when taking a lead. That way they are not at risk of being out on a drive that they fail to avoid.
This is actually a great question. Runners moving in foul territory are allowed right of way over the ball as long as they are in the their running lane.
Their running lane is where they are standing to a straight line to the next base.
So your average MLB player might lead off a good 5-8 feet off 3rd base line. They are allowed a path (the rule of thumb is about 3 feet each side of their current stance) to home plate.
Ball is hit directly at runner and he doesn't move. That is just a foul ball. Runner doesn't have to make a concerted effort to even move out of the way unless a fielder is trying to make a play.
Ball is hit (let's say highish or bloopish towards the runner). Runner sees 3rd baseman coming to make a play and stands there. He is out but for fielder interference.
Ball is hit right inside the line but runner is 8 feet outside line. He dives in front of the ball and is hit by it. The runner is again probably ruled out for fielder's interference, not being hit by ball.
Ball is hit 15 feet outside the line and runner is hit because he thinks it would be fun to get hit by ball. Well here we meet the definition of runner interference. However this is a hard call for an umpire to make if there was no opportunity of an out. I haven umpired and watched baseball for 30 years and have never seen anything close to this.