I agree that "dye an old pair of flats" is a piece of advice from a different era. The *only* cases I know of dying shoes are people who bought dyeable shoes specifically to have them dyed to match bridesmaid's dresses.
I'd be tempted to think of the request as a joke, because it's so bizarre. I'd also be tempted to give an advance present of nicely decorated wedding boxes for the bridal party to stand on.
Personally, if it were a casual acquaintance, I'd decline the invitation (or rescind the RSVP if the instructions came after I had replied). If it were someone close, like a family member, I'd go, but assume the shoe instructions were a joke.
Evil blarg would show up with flats and a bag over her head, and explain that she didn't want her ravishing beauty to make the wedding party feel self conscious.
On second thought, running shoes are flats, right?