I'm not sure if lives down means forgotten or that she lives in a floor down the protagonist:
"When your teardrops go sour and no longer fall
the splash cross the virgin that lives down your hall"
In this case, 'lives down' means a neighbor (in this case the virgin), probably within an appartment complex, whose door shares an internal hallway with your own. You would say that said neighbor 'lives down the hall' from you. You run into this phrase at work or schools as well when telling someone to go further 'down the hall' to find the door to their office or classroom. A hotel also would be a place where this phrase could apply as well.
You may also see 'a few doors down' or even possibly 'up the hall' to mean the same thing, though 'up the hall' feels to me like the direction specified is going the opposite direction than what is expected.
Ditto Michael Dorgan. I'd add that you can also say that someone lives "down the street", meaning he lives somewhere else on the same street. You can also say that you are "walking down the street", etc.
In general I think "down the ..." in this sense can be followed by any word that describes or refers to a line of things: a hall, a street, an aisle, etc.
Maybe not quite the same thing: People also say "up the river" and "down the river". In this case "up" and "down" have clearly distinct meanings: "down" is in the direction that the water flows; "up" is in the opposite direction.
The phrase "live down" is also an idiom meaning, "get people to stop thinking and talking about, or stop thinking badly of you for". As in, "Bob showed up for the staff meeting drunk! He's never going to live that down!"
But that's pretty clearly not the intended meaning in the example you gave. It doesn't make sense in context.
The "down" doesn't belong to "live" in this case, it belongs to "the hall". "Down the hall" means "further along the same hall", so on the same floor.