I was watching Mayday S11E02 about West Caribbean airways flight 708 and the pilots turned off the de-icing system partway through the flight. It's not why it crashed, but I was curious as to if there was any disadvantage to having the de-icing system on.
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.
It depends on the type of de-icing system. A jet like the MD-82 on that flight will use bleed air from the engines which supplies TAI (thermal anti ice) for engine nacelles and wings. That extra bleed air taken from the engines lowers their power output and decreases the fuel economy. If you are limited by EGT for available power, turning on anti-ice is going to degrade your climb performance. Likewise, on a descend using anti-ice might require running the engines higher than flight idle to supply enough air, which effects the descend path and lowers fuel economy.
So, on a jet with thermal anti-ice, it's not exactly dangerous to run the anti-ice when it's not needed, it just lowers your performance and economy a bit. If you on the other hand need every little bit of power you can get (following an engine failure for example) turning off anti-ice might give you just that bit of power.
On certain, mostly propeller powered aircraft, however there's a different system for de-icing (note that it's de-ice, not anti-ice) that works with inflatable boots on the leading edge of the wings and/or horizontal stabilizer. The idea is that you let a sheet of ice accumulate on the de-icing boots, and then inflate them using a compressor, which breaks the sheet of ice that is then blown away. Turing on the de-icer will normally cycle the compressor so it inflates and deflates the boots in regular intervals to break off accumulating ice. Turning on those de-icing boots too early will accomplish nothing, because the boots can only break off sufficiently hard ice. Some manuals even consider turning on the boots too early (less then 1/4" of ice accumulation) harmful, as the ice might not be hard enough to break off cleanly and instead stick to the boot. This advise is found in some older manuals, I do not have any hard evidence if this is still considered best practice or if it is an old wive's tale.