Is the A320 family allowed to take off with the flaps retracted?
For the A320 family it is not allowed. Above shows the takeoff configurations. Anything else and the takeoff warning will be triggered. I also checked the abnormal and special operations, nothing says a 0/0 takeoff is possible.

Each plane is certified for certain configurations. What applies to the A300 or 767 does not apply to all planes.

I took it one step further and checked the MEL (ignoring the fact that if the flaps are inoperative, the crew will be in for a treat when it comes to landing). At least one slats/flaps computer is required, and for the flaps/slats to be tested before the flight. The same MEL allows a takeoff if the landing gear can't be retracted.

(Click to enlarge.)



Is it impossible? Of course not. A 737-300 once crash landed on grass, was fixed, and took-off from that same grass field. Nothing is impossible. But for the day-to-day operations, the answer is no.