dropped infield fly

Yes, the runner can advance.

It doesn’t matter if it ball was dropped unintentionally or intentionally.  The infield fly rule applies:  the batter is out and the runners can advance at their own risk.  Even if the ball was dropped intentionally, the infield fly rule takes precedence over the intentionally-dropped airborne ball rule.  I think the reason for this precedence is that the umpire calls an infield fly before the ball is dropped.  Thus, at the time the ball is dropped, the batter is already out and the runners are no longer forced to their next bases.  Therefore, the rationale for the intentionally-dropped airborne ball rule doesn’t apply on an infield fly.

I hope that helps.

Agreed, ataha.  I’ve seen fielders intentionally drop an infield fly to try and coax unknowing runners into believing they have to advance due to the dropped ball.  But, as you say, once the infield fly rule is called, the batter is out and runners are not forced to vacate the base(s) they occupy at the time of the pitch.

The batter is out after the umpire calls infield fly, regardless of whether the ball is caught or not. The ball is still in play, and base runners may proceed at their own risk.