Supposedly this happened in MLB last year.
One out, men on first and third. Batter hits a long fly ball. Man on third doesn't tag up, he just trots home and into the dugout. Fielder makes a great catch (2nd out) and throws to first before the runner there can get back (third out). Runner on third never tagged up and the fielding team never appealed the play; they just trotted off the field. The run counted.
Now, the argument is: What would the ruling be if the first baseman then threw the ball to third as an appeal play. Run doesn't score even though the appeal comes after the third out of the inning? My take is that the run would not count.
One out, men on first and third. Batter hits a long fly ball. Man on third doesn't tag up, he just trots home and into the dugout. Fielder makes a great catch (2nd out) and throws to first before the runner there can get back (third out). Runner on third never tagged up and the fielding team never appealed the play; they just trotted off the field. The run counted.
Now, the argument is: What would the ruling be if the first baseman then threw the ball to third as an appeal play. Run doesn't score even though the appeal comes after the third out of the inning? My take is that the run would not count.