If there's a card that requires a card to be burnt in order to activate, and if I have no card to burn, I won't be able to activate the card. Does the card still get flipped (as the post-activation action) during the 'run your city' action, irrespective of whether the card was activated or not?!
When you run your city, you choose which stacks get activated. You are not compelled to activate every stack. So in your case, you wouldn’t activate that card and it wouldn’t get flipped.