Runner on first, a grounder is hit to first and the runner doesn't advance because he thinks the ball is going foul. The ball is played by the first baseman and ruled fair. The first baseman touches first base and then tags the runner, who is still standing on first base. Is this a double play or is the runner on first safe because he is no longer forced to advance to second by way of the batter-runner being out?
The runner on first is safe. As soon as the fielder stepped on first base, the force was removed, so the runner was perfectly entitled to stay at first base.
While you say this was "poor baserunning", the much more obvious mistake he was from the fielder; if they had simply tagged the runner before stepping on the base, it would have been a double play. The rule of thumb when getting double plays is simple: always get the lead runner first, as that's what keeps the force in effect.