Is a pullup resistor always required for a switch?

Ack, so I'm sitting here and another thought came up.

Do you need to worry about using pullup or pulldowns on outputs?

I have digital pins controlling a relay. Is it possible for random environmental fluctuations to cause a digital pin to output a high voltage and cause the relay to inadvertently actuate? Would it be necessary to add a resistor to make sure the voltage is always at 0V unless commanded to output?