TLDR: It's illegal if done with intent to render the bills "unfit to be reissued".
There's probably enough leeway there for the government to construe widespread defacement of bills with adverts for competing currency as illegal. I wouldn't try to start such a movement, more likely to attract the kind of attention bitcoin doesn't want.
Writing a web address is enough to render a bill unfit to be reissued? That's a stretch, IMO.
And it's a game to track bills not a movement. Bitcoin is not a currency.