Why do things stop accelerating? Before $t=0$: object is not moving
At $t=0$: I push object with sufficient force to overcome static friction
After $t=0$: Object accelerates. Resultant force is greater than $0$. Dynamic friction is less than the push. So why does the object stop accelerating after say $t=2$ (either starts to decelerates or reaches constant velocity)? $F_f = \mu N$ so it does not depend on the velocity of the object so as the object accelerates it, the push force and frictional force stay constant so the resultant force also does.
 A: Forces are interactions between two objects.  You push on an object, and the object pushes back with an equal but opposite force.  When the interaction stops, the force disappears.  No exception to this has ever been observed.  That is why it is called a law.
The object never stops accelerating as long as you keep pushing with a force greater than the friction force.  If you decrease your pushing force to be equal to the friction force it will move at constant velocity.  If your pushing force becomes less then the friction force, the object decelerates.  These are simple statements about how Newton's First and Second Laws apply.
A: 
the push force and frictional force stay constant so the resultant force also does.

Your scenario is not very explicitly stated, but I am assuming that you are talking about a human arm pushing an object over a rough surface.  If so, then the quoted statement is actually incorrect. The push force does not stay constant. As the push force decreases it becomes less than the friction force and the object decelerates.
The push force decreases because animal muscles and limbs produce a very non-constant force output. The force decreases with speed and also changes with joint angle. The force drops sharply to 0 as the arm becomes fully extended (joint angle reaches 180 degrees).
If you were to actually provide a constant push force then indeed the object would continue accelerating until other forces (e.g. air resistance) became non-negligible.
