Is there any disadvantage to having the de-icing system on?
The wikipedia article already linked in the question says:


  The system uses power from the engines, thus reducing their maximum performance. With the anti-ice system on, the highest altitude at which the overloaded aircraft could fly – without stalling – was reduced to 31,900 feet (9,700 m).


The de-icing system on most turbine aircraft (including MD-82 involved in that accident) uses bleed air from the engines, that is it extracts some air from behind the (low pressure stage of the) compressor. This air is therefore not ejected from the nozzle and not producing thrust, so the thrust is reduced. Or fuel flow rate increased if thrust levers are advanced to compensate, provided the engines are still below their maximum permitted thrust setting. Either way fuel consumption per mile increases.

Compressing air increases it's temperature, so this pressurized air is simply blown through tubes under the critical surfaces. There are other uses of bleed air, mainly pressurization and air-conditioning, but turning on deicing increases it's consumption significantly.

And note that this switching de-icing off and on was part of the accident sequence. It was only because they turned de-icing off that they were able to climb to FL330 in the first place.

It is also often not allowed to use de-icing when outside temperature is higher than some value because the heated surfaces could overheat.