I was listening to the Kinks’ “20th Century Man” and musing about how much I feel just like Ray Davies back in 1971 now as a “21st Century Man”. “Too much aggravation”, indeed…Anyway, I thought about how computers fueled that “aggravation” since then and I could’ve sworn that the Kinks tackled the subject early on, but after googling I found no Kinks song earlier than State Of Confusion’s “Definite Maybe” from 1983 mentioning computers. This led me to the OP’s question. I know a lot of songs and I have racked my brain for early examples of that sort, but I draw a blank. Dopers, help me.
This’ll get us started.
Stones, 1967, 2000 Man
*
“Well my wife still respects me
I really misused her
I am having an affair
With a random computer”*
From 1969 (so it doesn’t beat TreacherousCretin’s):
I’ve miles
And miles
Of files
Pretty files of your forefather’s fruit
And now to suit our
Great computer,
You’re magnetic ink.
This is what came into my head right away. Well, mine was this version 
It doesn’t actually contain the word “computer” (except as part of the acronym), but…
Sheldon Allman: “Univac and the Humanoid,” 1960.
Emerson, Lake & Palmer’s “Karn Evil 9” (1974) obviously isn’t the earliest, but it does feature dialogue between a human and a computer.
That’s a good one! I love the contrast between what could be called techno-babble for 1960 and the typically stern rendition in the folk song style of that time. Thanks for the link.
Not quite what the OP is looking for, but here’s a pop star interacting with a computer in 1966: Monkee vs Machine
The classic Gershwin/Duke song I Can’t Get Started was updated for Frank Sinatra’s 1959 version.
“I’ve flown around the world in a plane, designed the latest IBM brain.”
Not exactly a catchy tune.
Allan Sherman’s 1963 comedy song “Automation” is chock-full of references to computers (and the word computer).
“Stochatta” from 1960’s Music From Mathematics has a voiceover section (included in the track itself, at least on iTunes) in which the computer is mentioned—understandable, as that’s what most of the works on the album were performed with, or even composed by—if that’s close enough to qualify as a “song.”
1973 actually.
This isn’t a real match for the OP’s question, but the 1971 comedy album, *I Think We’re All Bozos On This Bus *by Firesign Theatre had a character try to disrupt the automatons at a Disney World-style amusement park.
Description Here.
“Clem and Barney join other tourists in various exhibits and rides, and eventually encounter a simulation of then-President Richard Nixon similar to the “audioanimatronic” President Lincoln at Disneyland. But instead of merely making a speech, it answers visitors’ questions with vague, positive-sounding replies only remotely related to the questions and completely unrelated to the citizens’ concerns. When Clem reaches the front of the line, he puts the President simulator into maintenance mode by saying, “This is worker speaking. Hello.” The computer responds with “Systat: uptime” and the length of time that it has been running. Clem then attempts to crash the system by confusing it with questions it can’t understand, or sometimes, even parse…”
It was an early pop-culture depiction of hacking techniques, rather than just a mention of computers. In 1971 I did not understand much of the story.
Does Daisy Bell count?
Computers were displayed in “This Island Earth” (1955) and “Forbidden Planet” (1956). In 1957, Spencer Tracy and Ktherine Hepburn starred in "Desk Set, in which computers were central. Tracy was the inventor of EMERAC (“Electromagnetic MEmory and Research Arithmetical Calculator”)
This might be the first one of these to be directly inspired by actual computers, as opposed to some purely imaginary mechanical monstrosity. Strange little funhouse mirror look at what’s now classic software, and some very influential pieces of software at that (such as ELIZA, the first great chatterbot).
I think that’s about par for the course with Firesign Theatre albums.
Back to the shadows again!
That was supposed to be Nixon? Huh. It sounded so much like Reagan that I thought the Firesign Theatre could see into the future.
So, why does the porridge bird lay his egg in the air?
I wrote a parody of “The Way We Were” in 1979 about TRS-80 programming. Does that count?
Bowie’s “TVC-15” was a full-length song about a computer, not just a throwaway line, but it was released in 1976, so would need a pretty big asterisk by it.
Anything by Kraftwerk in this oeuvre?
Were they singing about the computers in these movies?