I've shamelessly weighed in on everything else Taang related. So, I might as well add in my two cents here. ;D
Since taang is still a matter of "how" and what if" I offer several options depending on how their ship might happen.
"Bandages" by Hot Hot Heat:
At the moment I'd say it best exemplifies their relationship. They started out really rocky with a lot of petty arguements. And while they're still a couple of grumbling kids, they are healing up they're emotional rift well enough.
"Like a stone" By audioslave:
Not just for the obvious pun in the song title, but it would be a good consolatory song if they start to have feelings for eachother, revealed or not, and things turn bad, such as Toph being dragged back home to her parents, Aang being captured or going off to fight or study on his own for some reason, or god forbid.... kataang or soph shippers get there way!
"Too Bad about your Girl" By the Donnas:
If Taang starts out with Toph for some reason getting a crush on Aang, the rock beat and ego driven lyrics should work well for Toph's view of Katara's 'interference', if he still has a crush on her at the time. Boy I'd love to see and AMV about that.
"Gravity Girl" by Good Charlotte:
On the flipside, if Aang is the one who gets an unrequited crush on Toph, who might still be hung up on Sokka, the simplistic sound and near cheesy sincerity of this song seems to mesh well with Aang.
"Hot for Teacher" by Van Halen:
Once more, if Aang ends up with a crush for his earthbending teacher, this song is a must! Though Kataang shippers might also like it.
"The surd Song" by Bowling for Soup:
This works in one of two cases. Either, Aang developes a crush on Toph, but is too shy to say it, and she thinks he's still hung up on Katara and thus mistreats him out of bitterness until he eventually tells her the truth. Or, they just have a very angsty and argumenative relationship, which may actually be the case.
"Girl all the bad guys want" by Bowling for soup:
Ok, had to throw in one more 'if Aang crushed on Toph' song, and one more Bowling for Soup song at that. Something about their music has a very juvenile tone, that suits Taang well. Anyway, if Aang starts crushing on Toph, who has her warrior boy Sokka fixation, this song would be perfect, expecially if Aang started trying to act like Sokka to get her attention. How funny would that be?
"Underneath it all" by No doubt:
If by some completely unbelievable miracle, the super dooper scooby-doo Taang ending comes about, where they both end up in love with eachother. I'd say this song will do just fine, especially if done from Aangs point of view. Though it might suit Zutara fans better.
I figure I'll add in a couple of Azuki picks, in consideration of how I think that ship would go. I'd have to say either "Make me Bad" by Korn, or "Figured you out" by Nickleback.
And to show there are no hard feelings, let me give Kataang fans a helpful suggestion. "Aqueous Transmission" by Incubus. It has very oriental feel which fits well with the show. But also, the lyrics, heavily intoned with river mentionings and speaking leaving and returning to a lover should work well with Aang's failed attempt to free himself of the bond to Katara. Granted I'm not cheering for you. I just thought it would be spiteful if I didn't make the gesture.