Yes a runner must touch home plate before another runner is tagged out for the 3rd out of the inning in order for their run to count. This does not apply to force plays. If the final out is made on a force play (meaning they don't have to tag the runner but only the base to make the out). If the third out is made on a force play the run does not count regardless if the runner reached the plate before the out was made.

To further clarify, a force play is when a defensive out can be made by stepping on the base. This occurs when a runner is forced to try to advance to that base. So for example the batter running to first is always a force play as he must run to that base after hitting the ball. Also anytime there is a runner on the base behind you it is a force play at the next base because the runner is forced to try to advance by the runners running behind him. So if there is a runner at first and second there are force plays at third, second and first because the batter must run to first, which means the runner on first must run to second and the runner on second must run to third. However this only applies when the ball is hit on the ground or if there are 2 outs. If the ball is hit in the air the runners must stay on their bases to tag up after it is caught in order to advance. In that case there is no force play and they need to be tagged out.