Yes, a foul tip......is a live ball and the runner can advance.
You can find this in the rule book under the definition of foul tip.
NFHS Rule 2 section 16
A foul tip is a batted ball that goes directly to the catchers hands and is legally caught by any fielder. it shall be called a strike and the ball is in play (live)
Not sure of what level of play you are talking about, this is a commonly known rule at most levels....if this was a very young age league and the runner was lured away from his base by the actions of the defense or an unwitting umpire or coach, I can see leaving him at second (good opportunity to teach the players the difference between foul ball and foul tip).........but at any serious level of baseball, the players are expected to know the rules and if he left the base and was tagged the out would stand...