mlb rule

Score the run. The out at 2nd is an appeal, not a force.

See 4.09 (a) How a Team Scores

Yes it scores, it is a time play. The only thing that unscore the run would be a third out being a force or BR not making first. What you have is a live ball appeal but it looks like a force, it is not.
I am on the wrong computer to give you the exact rule cite but it under how a team scores which is 4.00 I think.
Here is a link to the rule, which as Dash noted is 4.09, make sure you read the last note. It is with first and third but is what covers your question.
4.09

It sure must have taken the left fielder a long time to get off the turf and get the ball to second base if he caught the ball diving toward the infield, and still couldn't get double the runner up before the runner on third scored.

I've seen this kind of play about 15-20 times and every time the back runner is all fouled up and not very fast. That's why he's takes off hoping to score from 2nd or 1st. I've seen two variations:

1)The outfielder gets up and rushes the throw and is off line to the bag. So the 2B in this play might have to come off a few steps and even block the ball and then steps on the bag. Many times the runner never returns to the bag.

2) He tries to throw to the plate and the ball is cut off and thrown behind the back runner trying to get back.