Hate to tell you this, but they really don’t mention Dumbledore’s scar at all after that scene. Therefore, I reserve the right to idly speculate:
There are two possibilities concerning the origin of his scar- magical or non-magical. I mean, we don’t even know that much. Was the scar an honorable wound gained in a fierce battle? Or did he have an accident involving one of those new-fangled muggle bikes?
If it was a non-magical accident, then I’m afraid the scar it stuck the way it is, doomed forever to be inaccurate.
If not, then I suppose it depends on the nature of the magic that inflicted the scar. Harry’s scar was only special because a bit of Voldemort’s soul happened to be in him at the time. If Dumbledore’s scar is shaped like the Underground, then basically it looks like a bunch of squiggly lines, right? I’m not exactly sure how a spell could scar you in such a way, as the only other instances of magic causing scars is when a body part is physically cursed off, like when George loses an ear. Most spells just happen to DO something, like make a tea cup dance or kill you. Unless the spell intentionally was cast to cause a cool scar (like a more concentrated, British version of Sectumsempra, which basically maims you), it would only do it’s desired effect. Maybe some magical animal scarred him, who knows. Either way, it is improbable that the scar was intentionally shaped like London underground, and as it wasn’t associated with it to begin with, it would have no reason to change to account for new routes or whatever. If it was intentionally cast that way, that really raises more questions than it answers, but frankly, if you can figure out how to make a spell that scars you with a map you should be able to make it update itself.
Actually, now that I think about it, that could apply to all other scar instances. As McGonagall asks if Dumbledore can remove the scar, we know it’s possible in some cases, so logically, if you can get rid of it entirely, why not change it? I mean, sure, it sounds painful, but it would only be a few tweaks, and hey, it’s a free map. Might as well.