Zatanna: Now, there's a man who looks
like he needs a little magic in his button-up life. Tense life?

Oliver: I wouldn't want to steal
the hocus-pocus away from the birthday girl. How about
a little something special for the someone
who truly deserves it?
Zatanna: Rough birthday, huh?
Chloe: You ever seen the napalm scene
in "apocalypse now"? More fire, less cake. Meaning she wants a more vivid birthday?
Buttoned-up doesn't mean tense so much as inhibited, conventional, careful about appearances. Think of a man carefully dressed in a neatly buttoned shirt, tie & jacket.
"More fire, less cake" - I would take this to be a reference to the birthday cake with its candles. Instead of a delicious cake with tiny dancing flames, this birthday is more like a napalm disaster. I am not sure here who is having the birthday - the girl, the button-up-life man, who? And are they seeking to reward or punish?
Chloe's having a birthday but everyone has
their appointments and leave the party.