Runner on 1st, gentle fly ball to second baseman...?

It would be called an infield fly.....so the batter would automatically be out and the runners on base don't have to advance.

While, in theory, if the second baseman drops the ball he could most likely get the runner at first base who is remaining there as to "tag up" after the fly ball, it is a highly unlikely occurrence. After the ball is stuck, the batter then hustles down to first base as to please his coach, and to not get doubled off. The majority of the time the ball has enough hang time for the runner to reach first and therefore eliminate the possibility of the double play. The average base runner runs down the line at approximately 3.9 seconds and even on a hot shot double play ball to a second base, short stop combination, it usually results in a bang bang play at first base. Any sort of hang time on the ball rules out the odds of turning a 4-6-3 double play. The most practical play that can come out of a play like this would be a base-runner trade. For instance if the current runner on first base was a fast runner and the batter was a slower runner, the second baseman could drop the ball and get the faster runner our at second, however, like I said before, it is extremely rare for the double play to work so most coaches teach to take the sure out and move on.

If the runner reached 2nd base safely, however overran it, then its now not a force out if he's tagged on his approach again to the base. That becomes very important if the tag-out is the 3rd out of the inning and there is a runner from third base who scored on the play. Considering it is not a drive out at 2nd, the run would rely as long as it scored earlier than the out was made. If you're without difficulty speaking a couple of 2nd baseman shedding the ball as he attempts to throw to 1st base for a double play, then its as much as the umpire's judgement related to whether or not he managed the ball to tag the base then lost it on on the transfer, or if he dropped it earlier than the out was ever made. If the umpire rules that the 2nd baseman managed the ball sufficient to file an out then it could be viewed a force out.

It the runner on first is faster than the batter, then it would be better for the 2B to drop the ball and then throw to 2nd for the force. The baserunner would most likely have been tagging up so they could easily get the out at 2nd.

Teams used to, but then officials created the infield fly rule. Any popup to an infielder with runners on first, first and second, first and third, or bases loaded and less than two outs is to be played as if the fielder caught it. The home plate umpire says "Infield fly!" and the batter is automatically out.

Ollie got it right. You figured it out, and so have the powers that be in baseball. They created the infield fly rule to guard against it. The ump will call the batter out before a catch is made based on this rule, to avoid the intentional drop.