How should this be ruled?

Question

Little league baseball. No outs runners on 2nd and 3rd. Batter hits fly ball to outfield that is caught, runner from 3rd doesn’t tag up and crosses home plate. Runner from 3rd heads into the dugout. Runner from 2nd tags up, advances to 3rd, then advances home safely due to an errant throw. Should the runner on 3rd be out once they enter the dugout or does an appeal need to take place. Also, is the runner on 2nd out for passing the runner on 3rd since they didn’t legally advance?

Answer 1

Runner on 2nd is definitely not out - they get to score. See for example this explanation in 5.09(b)(2) from the MLB rules:

This rule also covers the following and similar plays: Less than two out, score tied last of ninth inning, runner on first, batter hits a ball out of park for winning run, the runner on first passes second and thinking the home run automatically wins the game, cuts across diamond toward his bench as batter-runner circles bases. In this case, the base runner would be called out “for abandoning his effort to touch the next base” and batter-runner permitted to continue around bases to make his home run valid. If there are two out, home run would not count. See Rule 5.09(d).

Not an identical situation, but not all that different in that a runner legally scores after a runner does something illegal; the first runner's breaking the rules doesn't invalidate the second runner or cause them to "pass" them.

As for the runner on 3rd - yes, this is an appeal play. From 5.09(b)(5):

(5) He fails to retouch his base after a fair or foul ball is legally caught before he, or his base, is tagged by a fielder. He shall not be called out for failure to retouch his base after the first following pitch, or any play or attempted play. This is an appeal play;

So, the runner from 3rd is out, and the runner from 2nd scores legally, assuming the defense properly completes the appeal on the runner from 3rd.