When does a run not count on a fly ball?

Answer 1

When the runner doesn't properly 'tag-up' after the fly ball is caught.

In Baseball, to tag up is for a baserunner to retouch or remain on their starting base (the time-of-pitch base) until (after) the ball either lands in fair territory or is first touched by a fielder. By rule, baserunners must tag up when a fly ball is caught in flight by a fielder. After a legal tag up, runners are free to attempt to advance, even if the ball was caught in foul territory. On long fly ball outs, runners can often gain a base; when a runner scores by these means, this is called a sacrifice fly. On short fly balls, runners seldom attempt to advance after tagging up, due to the high risk of being thrown out.

When a base runner fails to tag up on a caught fly ball (for instance, if they started running too early, thinking the ball wouldn't be caught), they may be "doubled off", which results in them being called out. To double a runner off, a fielder must touch the runner's starting base while in possession of the ball before the runner returns to the base. If the baserunner appeared to tag up, but a fielder suspects the baserunner may have left the base too early (thus failing to legally tag up), the fielder may attempt to double the runner off by touching the runner's starting base while controlling the ball, before the next pitch is thrown. This is considered a type of appeal play. If the umpire agrees that the runner did not retouch after the ball was touched by a fielder, the umpire will call the runner out, and anything else the runner did during the play (such as score a run) is negated. Doubling a runner off is considered a "time play" (as opposed to a force play), which means that even if the doubling-off is the third out of an inning, any runs which score before the double-off will count (unless the run was scored by the same runner that was doubled off, in which case the run will not count in any situation).