What is the minimum thrust needed to takeoff?
You need at least enough thrust to keep the airplane flying at the speed of minimum drag. This is quite a bit higher than the minimum flight speed, so you should add something to quicken the acceleration to this point. Also, you want to climb eventually, so you better add some more thrust.

Normally, the static thrust of an airliner is at least a quarter of its weight. If the airliner is empty, this can become as much as half of the weight.

One reason is altitude capability: Since thrust goes down with the density of air, the static thrust in the cruise altitude is only a quarter of sea level thrust. The thrust of a modern high-bypass-ratio engine drops with speed, so at cruise speed and altitude, thrust is roughly a sixth of sea level static.

The second reason is safety: The take-off should be continued even after one engine fails in the late acceleration phase. Now a normally two-engined plane has only half as much thrust available and should still get into the air, so it doesn't crash into whatever follows at the end of the runway.

The thrust needed to sustain flight is about 1/18 of aircraft weight, and if you factor in the multiples given above, you will notice that if the aircraft can fly at full thrust in cruise, this fits nicely with a static sea level thrust force equivalent to one third of its weight force.

The factors for take-off thrust are:


Runway length: Short runways need more excess thrust for faster acceleration.
Runway elevation: Higher places have less dense air, so more speed is needed to get airborne, and less thrust is available from the engines than at sea level. 
Runway slope. Taking off downhill is equivalent to having a little extra thrust.
Air temperature: Colder air is denser, so minimum speed is lower and the engines develop more thrust.
Wind speed: A headwind is equivalent to starting the takeoff run at this speed.
Minimum flight speed: A low wing loading and low-drag flap settings reduce the speed at which the aircraft can take off.