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.