When someone looks at your lips, does it mean they want to kiss you?

Seems like you just answered your own question bucko.  ;-)

That, or it could be that throbbing, shiny cold sore.

It would depend upon many factors, such as what the situation is, who the person is, context, etc.  
Just the other day, my barber had to look at my lips while shaving my moustache. Had she not looked at them, I hate to think what kind of jagged scar that straight razor would have left on me if she slipped up. 

No. Its usually because i need chap stick.

it might

Mine are hidden with fur.

Absolutely!! Remember that, D. Every single time someone looks at your lips, you should close the distance by leaning in, and pucker up for good smooch. 

I don't think I'm quite astute enough to notice when someone is specifically looking at my lips. I'd assume they are getting ready to compliment me on how luscious they are, though ;)

That's probably why...

Yes. what else can it mean.

I could also mean that your lipstick was not put on right or that yout have food stuck on your lips. :P