Yes.  2 runners cannot occupy the same base at the same time.  Since runner on 3rd had advanced beyond 3rd base, and runner on 2nd advanced to 3rd, therefore becoming the legitimate 3rd base occupant, lead base runner is out.  No tag is necessary.

Okay, Lou_B,

Good call.  I did a search and under section 7.03 of "The Official Rules of Baseball", it appears that you are correct.

Official Rules: 7.00 The Runner

7.01

A runner acquires the right to an unoccupied base when he touches it before he is out. He is then entitled to it until he is put out, or forced to vacate it for another runner legally entitled to that base. If a runner legally acquires title to a base, and the pitcher assumes his pitching position, the runner may not return to a previously occupied base.

7.02

In advancing, a runner shall touch first, second, third and home base in order. If forced to return, he shall retouch all bases in reverse order, unless the ball is dead under any provision of Rule 5.09. In such cases, the runner may go directly to his original base.

7.03

Two runners may not occupy a base, but if, while the ball is alive, two runners are touching a base, the following runner shall be out when tagged. The preceding runner is entitled to the base.

I thought I'd give it a shot without referring to the official rules, just to see if what I thought I know was correct.  It wasn't.   Good job to you.

Maybe next time, I'll get it right.

Mike