I'm trying to locate a specific episode of the Original Star Trek with William Shatner as Captain Kirk. It is about 3 or 4 members of the crew who beam down to a planet and find giant robots battling each other. With investigation they discover two teenagers, each in a separate building, who have never seen another human being in their lives, not even each other. They are the only remaining humans on the planet.
This is a real long shot because the timing on when it aired would not at all match the Star Trek-TOS timeframe. But, I believe rather than Star Trek I believe you are referring to the SeaQuest episode Playtime
Points that match include:
There only being two teenaged humans left on Earth.
"A distant call for help pulls the SeaQuest through an underwater anomaly, propelling the boat 225 years into the future where they find that mankind has destroyed itself, leaving only two teenaged kids as the only Human life left on Earth."
The title "Playtime."
Because they have never actually met each other, the two teenagers remotely control robots to battle each other in a continuous playing mode.
As commented by a viewer on the IMDB site:
"The first 15 minutes of this episode are OK, the time travel, the remains of human society, a supercomputer and mechwarrior-like robots fighting (although those were cheap CGI-effects, but still worth a watch)."
A partial match that you may be mis-remembering, rather than beaming down, the crew actually travels through a black hole in time to reach them.