Effect of zeroes in system stability
The stability depends on both the poles and the zeros of the plant and controller.

To see this consider the asymptotes of the diagram of the loop gain/phase. Imagine moving the zeros changes the phase at crossover and therefore affects the stability.

Looking at it more formally, the dynamic characteristics and stability are determined by the poles of the closed loop transfer function.

Consider a control system as shown below.



Let the plant be represented as a transfer function of poles and zeros $$G=B/A$$
and let the Feedback (or controller) be represented as a transfer function of poles and zeros $$H=S/R$$

The closed loop transfer function is therefore:
$$\dfrac{B/A}{1+\dfrac{BS}{AR}}$$
Where the demoniator is the characteristic equation which determines stability. Simplifying:
$$\dfrac{BR}{AR+BS}$$

So the stability is determined by the characteristic equation (poles):
$$AR+BS=0$$

Therefore the stability depends on the poles and the zeros.