Are there telephone touch-tones in there, or is that just an artifact of whatever primitive synthesizer they were using?
You’ll be held fully responsible if I get a “Don’t Call Us, We’ll Call You” earworm out of this.
Not an answer, but — recently I posted a Youtube link to the song for one of MeanMrMustard’s alphabetical song polls (maybe the “O” one ), and was depressed by the audio quality of the 3 YT versions I checked out.
Maybe there’s a quality version on YT, but listening to physical media or a high quality digital file might help when answering the OP’s question. Always loved the song.
I just heard them. They start about 3:11 into the song.
To the OP: Might help if you posted a link to the song.
Listening to the end of that, I’m not hearing ‘telephone touch-tone’ sounds, but I do hear some sort of ‘clinking’ sound that may or may not be what the OP is referring to. To me, it sounds like some sort of lo-fi percussive effect, like tapping a drumstick against a muted cowbell or something.
ETA:
I just tried listening to the song on my Amazon Music account, which is listed as ‘Ultra HD’ at a loudish volume, and I still do not hear anything that could be considered touchtone sounds other than even more clearly the ‘clinking’ I mentioned, which again just sounds like a percussive effect. Like tapping a drumstick against something. A drinking glass maybe?
Am I missing it? I recently tested my hearing, which is still working well in higher frequencies for my age.
Thanks for the link, however, the mix sounds like mud through my headphones. The finger-picked guitar notes are borderline inaudible except at their loudest, and the bongo-type drum is muffled.
The voices are very clear.
I think the best way to describe what I’m hearing is flat, high notes with absolutely no modulation and the envelope of sound lacks any ramp up or down. It’s like someone flicking a switch; the sounds turns on, then off, then on and off again. Also, they’re all the same note. There are four of them, in two groups of two, each sound about 1 second long.
I’m pretty sure it’s just Christine ( 
) playing a note on whatever keyboard she’s playing.
Turns out I’ve got a Fleetwood Mac Greatest Hits (1988) CD on the shelf. Imported it and listened through mediocre headphones.
Heard the tone at the 1:34 mark. After that it appeared at 1:59, then 2:13. After which it arrived about every ten seconds (often as a pair 3 seconds apart). So… approximately:
2:13 and 2:16
2:23, 2:32
2:42 and 2:45
2:51 and 2:54
and then in pairs to the fade out.
Yeah, it seems like a single note on some type of keyboard.
I listened to a couple of different versions with my Sennheiser HD 6XXs and my Moondrop Arias. I don’t hear what I would call “touch-tone” sounds but I do hear some guitar harmonics in places–probably not what the OP is hearing.
Not hearing it either.
Thank for the time stamps. I hear it, but it doesn’t seem to serve the song in any way. Could it be some sort of cue that should have been removed from the final mix?
Same. I think they are gaslighting us. 
Hello. So I cut off the first minute of the tune and posted it on YouTube. With headphones I can hear (pretty easily) the first Tone at 11 seconds (maybe 10). The other tones are indicated in the comment box of the video.
Up To My Beeping Navel Video
Ah, I can hear it now (shitty speaker bar didn’t help). They are about a quarter note in length, and I do hear the resemblance to telephone touch tone. Like others said, not sure why they are there since they don’t seem to be integral to the song.