In addition to the previous answers, you could argue that yes: provided you had a long enough runway you could theoretically perform a takeoff run that accelerates you to a speed which is sufficient for taking off and remain airborne with no flaps (and for this exercise let's assume an A320 has a flaps-up speed of 210 knots, give or take), but you'd be failing to include very important aspects in your analysis:
First, as you move faster, the amount of runway you are "consuming" increases at an ever growing rate. A 50 knot increase in groundspeed when you are rolling at 80 knots consumes far less runway (as in several thousand feet less) than a theoretical increase of 50 knots when you're travelling at 160 knots (even if acceleration was linear, which it's not, and it took the same amount of time. So, the longest runways that you generally find in commercial airports (around 14-15 thousand feet) would probably need to be in the vicinity of 40-50 thousand feet to takeoff with no flaps in a loaded A320.
But, second: landing gear are not designed and built to withstand the stresses of such speeds. They'd probably wear down or even be damaged/destroyed in a single takeoff run and likely EverbodyDies™
Also, third (and probably most important), aircraft are built specifically to minimize the amount of energy required to put them in the air and physics, being the crazy rascal it tends to be, will find a way to make this even more complicated by means of the ground effect. As you travel faster through the ground on a vehicle which is designed to fly and generate lift (or, seen in the opposite way, to generate as little downforce as possible) would tend to generate enough lift to get separated from the ground before actually achieving the desired flaps-up speed and this would (and has been) fatal: the aircraft would initially climb a bit, maybe 50ft and then lose the lift provided by the ground effect ... You'd be left flying insanely fast (let's say, for the sake of argument, at 180 knots), 50ft above the ground and suddenly without the ability to generate enough lift to remain airborne (because you are now too far from the ground for ground effect to exercise any influence over your aircraft). You now plummet into the ground like a piano, destroy your landing gear and severely damage your aircraft's structure, break up and fire up in a massive ball of flames at 180 knots (over 200 miles an hour).
And yes, this has happened: 20 years ago, an Argentine Boeing 737 started its takeoff run with no flaps and killed 65 people when it fell back onto the runway after losing the small amount of lift provided by the ground effect... You can read more about that accident HERE.