This had made me wonder for quite a while, so
I poked about a bit and found my own answer.
Yes, they do. In certain states, a phone call can be legally recorded so long as one party consents. Crank Yanker does their calls from Las Vegas, an incentive for the comedians and guest starts to have a gambling junket while making some crank calls.
But they do have to get permission from the other party before they air the clip,
According to Daniel Kellison, one of the shows producers:
“If making the calls sounds like a monumental task, getting permission from the victims to air them is even harder. After all, not everyone who is pranked laughs about it afterwards. “We get clearances on all the calls we put to air,” Kellison says, “and we have an excellent clearance department . . . at least 90 per cent. I am very, very persistent,” he laughs.”
When I went to get the link I took that quote from, it was a log in page… Funny, it let me read the article, but any further trys, I just get the log in page.