If the response was flat, a zero will cause the response to rise at 6 dB per octave above the zero frequency. If the response was rising at 6dB per octave, a zero will cause the response to rise at 12 dB per octave above the zero frequency. If the response was falling at 6dB per octave, a zero will cause the response to flatten above the zero frequency. That is, it is essentially impossible to get a free zero. In this case, we must first accept 14 dB of overall loss below the zero.
Poles CAN occur at zero frequency.
One example is the phase comparator of a phase locked loop. A pole at DC is called a true integrator. A zero may also be called a differentiator. However, since you can't buy a zero without buying a pole, there are no true differentiators. Those of you who have taken calculus can appreciate this.