Does the device/charger consume power after charging is complete?
Like any other device, regardless if the battery is full or need charging it consumes energy all the time. When your device is fully charged and still plugged in then the charger replenishes the energy used by the phone. It uses less energy than when charging but still. This is called Trickle charging
It is advised to unplug device because keeping you battery fully charged all the time will reduce its capacity and make your battery die faster. Battery needs to be partially discharged and then be fully charged on a semi-regular basis in order to prolong its life.

The following sentence is from  Article on prolonging battery life


  Let’s look at real-life situations and examine what stresses lithium-ion batteries encounter. Most packs last three to five years. Environmental conditions, and not cycling alone, are a key ingredient to longevity, and the worst situation is keeping a fully charged battery at elevated temperatures. This is the case when running a laptop off the power grid. Under these conditions, a battery will typically last for about two years, whether cycled or not. The pack does not die suddenly but will give lower runtimes with aging.


You can also read about partial discharging and charging here:

Techrepublic - Five tips for extending lithium-ion battery life