Up to what thrust setting of a gas turbine engine can the airplane stay stationary on ground? I noticed that when a jet engine airplane is started or is conducting run-up, the airplane remains stationary even though the engine is already burning gas and is supposedly producing thrust. 

What keeps the aircraft stationary while the engine is spinning? Up to how much percentage of the engine's maximum thrust can you put the thrust setting on as to not have the aircraft move about its stationary position?

An alternate question would be: "On what thrust setting on each engine (assuming twin engine) will the airplane start to move and accelerate?"
There are two answers to this question depending on how you look at it. 

If the brakes are on: Full Thrust wont even move the plane this is a requirement of the braking system. 

Brakes off: Somewhere a bit above idle. John covers it in his answer nicely. It will vary by loading and aircraft. 

A propeller plane can get moving even at idle thrust. While jets don't do run ups (unless required for maintenance), propeller planes preform a run-up prior to takeoff where the brakes are utilized to hold the plane in place at a high thrust. A proper short field take off will also involve the application of brakes and full thrust.