I'm curious how tracers work exactly.  I found a few old 7.62 tracers in an ammo bin, and almost all the orange paint has been worn off.  This paint doesn't do anything other than differentiate the tracer from a regular 7.62, does it?  I mean, it has nothing to do with the bullet lighting up, right?

I assume not, so how does a tracer work?  Where exactly is the tracer material (phophorus?)?  How does it become visible?  How does it ignite?

the paint has nothing to do with a tracer, other than indicating that it's a tracer.
the tracer chemical is at the base of the bullet and is encased in a very thin metal (sometimes plastic) case, this burns through and starts the chemical burning, preventing it from igniting in your barrel. different chemicals produce different traces. the chemical is ignited when the flash and hot gasses penetrate said casing.