Which episode of Johnny Carson did he ask John Denver to leave the show?

There is no such episode. The fictional story is a common addition to another more popular urban legend claiming some famous musicians ordered all audience members of a particular religious or national group to leave their concert.

Below is a twist (collected via email) telling how Johnny Carson asked John Denver to leave his show:

I was once told that John Denver (folk singer of songs like Rocky Mountain High and such) was on the Tonight Show one night. He was a huge patriot and apparently had no love whatsoever for Jehovah’s Witnesses. Apparently, as the story goes, John gets up to sing a patriotic song and before he begins, tells the audience, “If there are any Jehovah’s Witnesses in the audience, you may want to leave right now.” And then he sings his song. When he’s done and goes to the couch to chat with Johnny, Carson apparently says, “I want you to know something. You see that camera guy? He’s a Jehovah’s Witness, and so is that lighting guy and so are a bunch of people on my staff. I hire them because they are honest and hardworking. And now, I’d like YOU to leave.”

According to internet myth-buster Snopes.com, the legend about the hate-filled musician who orders members of particular groups to leave his concerts has over the years been attached to a number of performers, including Clint Black, Phil Collins, John Denver, Gloria Estefan, Don Ho, and Keith Urban. Usually, the music lovers being ostracized are Jehovah’s Witnesses, but when the rumor attaches to Phil Collins, the ones being driven away are Jews, and when it is told of Keith Urban, they are Canadians.

Also, in those latter two cases, putative reasons for the banishment are offered: Collins is said to be a supporter of the Palestine Liberation Organization (PLO) and so regards Jews with disfavor, and Urban is said to be down on Canadians for their country’s not supporting the U.S. in its war in Iraq.

The most common version of the story has John Denver, he of the angelic face and kindly manner, demanding that Jehovah’s Witnesses in his audiences depart. No reason is given for his enmity; no explanation offered as to what they might have done to incur his wrath. He orders them away, and (we are left to surmise) they meekly get up and leave. Read more here.