The 7th, 16th and 17th, all very residential areas, are very quiet after midnight, or even before. t
You'd have to go to the Champs-Elysées and/or around Les Halles (Au Pied de Cochon, Chez Denise) for find restos open that late and 24/24.
You could walk to L'Alsace, 39, Avenue des Champs-Elysées (open 24/24h) and have a snack (quiche or whatever). Do not expect high end food though:
You'll find some more suggestions here: