According to Newton's thirdlaw, every action has an equal and opposite reaction. When a comb is charged and held near small pieces of paper, the comb exerts an electric force on the paper pieces and pulls them toward it. Why don't you observe the comb moving toward the paper pieces as well

Due to the fact that the comb is in terms of mass bigger than pieces of paper

Explanation:

In terms of mass, the comb is way too bigger than the piece of paper so it is virtually impossible for it to move towards the pieces of paper

According to Newton third law action and reaction are equal and opposite

Mathematically

P=m*v

Where

P = impulse is Newton

m=mass of the body

V = velocity of the body

Now since the comb has a greater mass it will not move to the paper, else the paper must have to move to it.