Does the Runner Score

Run scored by runner who was originally on third base counts.

The appeal for the out at second base does not affect the status of a preceeding runner who scored before the appeal.

Blown call, run should count. Appeal is a "time" play, and if run crosses before third out is made it counts. This is a common point of confusion as many think the runner going back to tag up is a force play, but it is not.

Absolutely, run should count

In fact, a little baseball trivia:
You may need to take the "fourth" out on a tag play.
Let's say you have runners at second and third with one out.Ball is hit in the gap to right center but the fielder makes a spectacular dive for out two.
Both R2 and R3 take off on the hit thinking it will fall in. R3 crosses home.
Ball is thrown in to second who tags the base and the ump gives the out call for the third out.
The defense runs in.

Run Counts!

Unless the defense ALSO appeals the runner at third leaving early before retiring from the field, the run counts because he crossed the plate before the third out was called.

The Houston Astos gave an extra run to the Phillies this way in an pennant game in 1980.

As noted above, the tag is a timing play, not a force.

All true, but he said they already tried the appeal on the runner from third, and he was ruled safe.