With a man on third and second, batter flies out to right and is caught for the second out. The runner on third tags up legally. The runner on second left early, right field throws to third to get the runner on second going to third. He left early they throw to second and get the third out. Does the run from third count?
The run would count, so long as it was the preceding runner (meaning, the one further ahead on the bases - in your case, the runner on third base). See rule 5.09 (d) in the MLB Rules:
(d) Effect of Preceding Runner’s Failure to Touch a Base Unless two are out, the status of a following runner is not affected by a preceding runner’s failure to touch or retouch a base. If, upon appeal, the preceding runner is the third out, no runners following him shall score. If such third out is the result of a force play, neither preceding nor following runners shall score.
5.08 (a) also makes this clear:
(a) One run shall be scored each time a runner legally advances to and touches first, second, third and home base before three men are put out to end the inning. EXCEPTION: A run is not scored if the runner ad - vances to home base during a play in which the third out is made (1) by the batter-runner before he touches first base; (2) by any runner being forced out; or (3) by a preceding runner who is declared out because he failed to touch one of the bases.
If the runner on third base did not tag, and then both runners scored, then the run from the runner on 2nd would not count if the third base runner's out was the third out in the half inning. However, the second base runner does not impact the third base runner, unless he is forced out before the runner on third base scores.