No, it's not necessary. There are advantages and disadvantges, depending on whether you have live plants or a filter providing water movement.
The traditional arguement for air pumps is that they provide agitation at the water surface which increases oxygen exchange > providing more oxygen for the fish.
In a planted tank the agitation of the water surface by an air pump also allows C02 (produced by the fish) to travel more easily from the water into the air. As C02 is needed by the plants for growth, this could potentially result in reduced plant growth, if C02 is the limiting factor (which is a whole other discussion). Therefore it's generally advised that air pumps aren't needed in planted tanks.
If the tank does not have filter or powerhead providing water movement, an air pump can be advantageous because it mixes the tank water around better. This means the temperature is more even throught the tank and nutrients are more evenly distributed around the tank.