Having a treadmill runway would be the same situation as if you had a tailwind that equaled that of your groundspeed at takeoff. The net horizontal air movement would be zero, and since the lift from the wings by air rushing over it is the only force in the upward direction the net upward force would be zero.
This also depends on what kind of plane we are talking about. VTOL aircraft obviously have air momentum flows downward causing an upward resultant force, but other aircraft such as the YC-14 direct portions of the engine exhaust to flow over the wings thus assisting takeoff by providing high airflow over the wings causing a greater lift force than just takeoff speed air.
Aircraft with strong tailwinds need greater groundspeed to takeoff which translates into a longer runway run. And if the tailwind was as fast as the normal takeoff airspeed then the plane would need to double its groundspeed to takeoff, if the tailwind airspeed increased proportionally (faster you go faster it would blow) to the aircrafts relative speed then it would never achieve the lift force it would need to become airborne and thus would not take off.
In the reverse, this is why headwinds are great for landings and takeoffs because the groundspeed needed for appropriate lift is smaller and safer.
