Sometimes when listening to someone talk in quiet conditions (i.e. no background noise - particularly on headphones), I can hear and follow what sounds like harmonic overtones in the voice. In other words, I can hear harmonics which sound like they're related to the fundamental. If I concentrate on them the voice starts to take on this sort of metallic, almost phasey quality. I used to notice this really strongly when I was a small kid, in fact I have very vivid memories of noticing those overtones when listening to people speak on TV at about age 3. Of course I had no idea what they were. Perhaps I just ended up filtering them out as I got older, but recently, perhaps because of the extra attention I've been paying to sound since getting into audio production, I've begun to hear them again. Especially when I'm watching a tutorial video with headphones on. Is it that everyone can hear them but doesn't pay any attention to them, or am I just weird? 

Can't say I've noticed that myself, but isn't there a singing technique that uses harmonic overtones?

Yes overtone singing, it almost sounds a little like that but obviously not as strong. 

The overtones are always there, most of the time the brain interprets the full complement as the sound of the voice, but perhaps you are talented or unhinged in a way that focuses on single pitch regions. 
 
http://blogs.scientificamerican.com/roots-of-unity/your-telephone-is-lying-to-you-about-sounds/
 

Try this on for size!  I swear she sounds like a Theremin...
 

Anna is hot ...
 
James are you sure you're not experiencing some sort of aural flashback?

My dog says she can hear that kind of stuff.
 

sharke
Sometimes when listening to someone talk in quiet conditions (i.e. no background noise - particularly on headphones), I can hear and follow what sounds like harmonic overtones in the voice. In other words, I can hear harmonics which sound like they're related to the fundamental....
 
Is it that everyone can hear them but doesn't pay any attention to them, or am I just weird? 

Yes - there are overtones... most people just don't pay attention to it - it's just what a voice sounds like to them. I've always heard this... yet people look at me like I have a third eye when I mention it. Some people can hear the nuance... some people just know it as "that's a pleasant voice", or "that's an annoying voice"... they don't know (or care) why.

Tuvan throat singing. Quite mesmerizing.

I think some voices have it more than others for sure. The best way to describe the phasey sound I hear is to say that it's a little like Frank Zappa's voice of The Central Scrutinizer on the Joe's Garage album. Obviously not as strong as that, but like a little of that effect blended in. It's almost like this ghostly voice I hear on top of the normal voice.