“Don’t touch that dial!” is said going into a radio or television commercial (or the next show), telling the listener or viewer to stay tuned to the station. “Ah-ah-ah! Don’t touch that dial!” has been cited in print since at least 1939.
 
A similar broadcasting saying is “Stay tuned!”
 
       
11 September 1939, Cleveland (OH) Plain Dealer, “Main Street Meditations” by Eleanor Clarage, pg. 7, col. 4:
About that time they decided they had had enough war bulletins for a while. The husband reached over to switch off the radio, but just as his hand touched it, a voice commanded him: “Ah-ah-ah! Don’t touch that dial!”

It was merely the announcer’s way of telling listeners to stay tuned in for the next program, but coming on top of their conversation, it was a bit startling.
   
27 March 1941, San Antonio (TX) Light, pg. 10B, col. 3 ad:
No! No! No! No!
Don’t touch that Dial
KTSA
First in San Antonio!
 
5 March 1948, Advocate (Baton Rouge, LA), pg. 8B, col. 4 ad:
DON’T TOUCH THAT DIAL
IT’S FRIDAY NIGHT ON WLCS
Rivet your dial to your local ABC Station every Friday Night…and you’re set for an evening of grand entertainment.
 
Google News Archive
26 August 1953, Miami (FL) News, pg. 8A, col. 4:
DON’T TOUCH THAT DIAL
—IT MAY BE COLOR TV
Associated Press
 
Google Books
January 1958, Boys’ Life, pg. 26, col. 2:
Ah, ah, ah . . . don’t touch that dial ... we suggest you sit through this because we may still be able to contact Palomar Observatory ... in which case we will interrupt Captain Moonlight.
 
OCLC WorldCat record
Don’t touch that dial
Author: U.S. Radio Band.
Publisher: Los Angeles : ABC, 1976.
Edition/Format:  Music LP : Rock music : English
 
OCLC WorldCat record
Don’t touch that dial! : Radio programming in American life, 1920-1960
Author: J Fred MacDonald
Publisher: Chicago : Nelson-Hall, ©1979.
Edition/Format:  Book : English
 
Wikipedia: KOME
KOME was a radio station in San Jose, California, heard at 98.5 FM from 1971 thru 1998. Currently, the 98.5 FM frequency is home to KUFX (“K-Fox”).
(...)
Several on-air slogans suggested the sexual connotations of its call letters, probably the most memorable being Chevy Chase exclaiming “Don’t touch that dial, it’s got KOME on it!,” as well as others saying things like “K-O-M-E all over your radio dial,” “Wake up with KOME in your ear,” and “The KOME spot on your dial.”
 
OCLC WorldCat record
Don’t touch that dial! : a TV/commercial themes medley : for SATB voices with piano, bass and percussion
Author: Mark A Brymer
Publisher: Milwaukee, Wis. : H. Leonard, ©1985.
Edition/Format:  Musical score : Popular music : English
 
OCLC WorldCat record
Don’t touch that dial : the impact of the media on children and the family
Author: Barbara Hattemer; H Robert Showers
Publisher: Lafayette, La. : Huntington House Publishers, ©1993.
Edition/Format:  Book : English
 
OCLC WorldCat record
Don’t touch that dial : hits n memories of Australian radio
Author: Wayne Mac
Publisher: Canberra : W. McCardell, 2005.
Edition/Format:  Book : English