Yes, if it's caught for an out. I saw Jose Canseco make a smart play once (which surprised me because he was usually pretty dumb.) Score tied in the bottom of the ninth, one out and a runner on third, Canseco was playing right field. There was a long fly ball hit towards him, and he let it drop foul rather than catch it for the out--because the runner on third would have tagged up and scored the winning run.
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Yes. A runner may tagup and attempt to advance after any fly ball is caught, regardless of whether it is fair or foul.
If a foul ball is not caught in the air, the ball is dead, so there would be no tag up opportunity. If it is caught in the air, the runners can tag up and advance at their own risk.
If the ball is held long enough to have possession and called an out then yes. If the ball is not held long enough for an out to be called then no.