This song is about Fight Club. If you don't believe me, I've taken the song and managed to connect it to the movie (and since the movie possess many of the themes talked about here that was quite easy).
"Lately, I've been, I've been losing sleep, dreaming about the things that we could be"
The narrator (Edward Norton) is clearly losing sleep dreaming about what his club could become.
"But baby, I've been, I've been praying hard,
Said no more counting dollars, we'll be counting stars"
He is sick of society telling him to earn a living, working at a crappy job to simply buy stupid crap he doesn't need. He'd rather be knocked out and counting stars (you know in cartoons when a character gets hit on the head and stars start swirling around them. That's what I mean) by fighting then wasting his life.
"I see this life, like a swinging vine, swing my heart across the line"
This line is saying that he sees life like a potential thrill ride, and he wants to swing on it as much as possible. The swing my heart across the line is saying that he doesn't care if he steps out of line, it's part of the fun.
"In my face is flashing signs, seek them out and you shall find"
This could mean that it's clear he's been in a fight from all the blood and wounds on his face,
Or
It could mean that he's clearly done something exhilarating because he's flashing signs such as smiling and such.
"Old, but I'm not that old, young, but I'm not that bold"
This might be saying that he hasn't doesn't done much in his life, or he isn't that bold. It might be in here because he wants to make an impact on the world in some way.
"I don't think the world is sold, I'm just doing what we're told"
I think this is him thinking that the world doesn't really know what to do, where it's going. So he might be sick if this fact, and sick of following orders from others trying to advance a society with no purpose, no meaning. That is probably what drive him to make the Fight Club, sick of being in a pointless society with no freedoms.
"I feel something so right, by doing the wrong thing,
And I feel something so wrong by doing the right thing"
He feels something right by breaking the set rules of society, and feels stripped of his freedoms and himself doing what society perceives the right thing to do.
"I could lie, could lie, could lie"
He is sick of faking himself. From destroying what he is and covering it up with what you see. He is sick of lying to the world about what he thinks is right, and lying to himself about who he is.
"But everything that kills me makes me feel alive"
Literally, getting his ass handed to him in the movie seems to be exhilarating to him. The fact that he accepted the truth about himself, rather than the lie, has made him feel alive.
[I will skip the parts I have already done to avoid repeating myself]
"I feel the love, and I feel it burn, down this river, every turn"
He has a love for breaking the rules, feeling alive, being free. Like a fire that keeps burning brighter and bright (providing it isn't a contained one that will eventually burn itself out)
The river refers to life, and he wants to be completely free. He has set himself free from societies chains, but he is still in it's cage.
"Hope is a four letter word"
I think he has hopes for what he is doing and that it will succeed.
"Take that money, watch it burn"
He wants ignore what he has been raised to be. To forget about what society has taught him and let freedom ring. To take what's left of society and watch it burn. To simply forget about all of the things society has made looks important and worth fighting for, in this case money, and experience happiness.
[Skipping ahead again]
"Take that money, watch it burn,
Sink into the river of the lessons I learned"
This is about letting go of society and burning it, and then thinking about it, or sinking into a river of the lessons you'ved learned. Reflecting on what you've done, how you'ved changed, what you'ved changed.