Is it possible to quantify electric power consumption on a fuse?
A fuse is basically a resistor where resistance varies from almost zero to infinity.  So just like a resistor, measure the voltage drop across the fuse, and the current through it.  Calculate the Volts times Amps and you'll get Watts.  Since the resistance is almost zero, the voltage drop will also be almost zero.  And so the watts will be almost zero.  Not exactly zero, but close. 

When the fuse is blown, the resistance is near infinity.  This will make the current basically zero.  And thus the watts will be basically zero.  

The interesting part is when measuring the power while the fuse is blowing.  It's not impossible, but not easy either.  Basically you work out a system where you can graph the current and voltage on an o-scope then do some tricky math to work out the amount of energy required during the fuse-blowing event.  I've never done this, so I couldn't tell you what it looks like.