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.
I don't think you can count on the fuse's resistance to be stable or repeatable. Put a small, but known, resistance in series with the fuse (0.1 Ω), and measure the voltage across it. The voltage divided by the resistance will tell you the current. V / R = I
Maybe there are special components that act both as current-sensing resistors and as fuses, but I don't know of them.
Simultaneously, measure the voltage across the thing that's supplying this current (like the voltage from the fuse to your circuit's ground, if the fuse is on the high side).
Multiply current by voltage to get power. P = V · I
The power consumption of the fuse itself: $P_{FUSE} = R_{FUSE} * I^2$
The contribution of a fuse's power dissipation becomes significant when you count all the watts and milliwatts on the way to finding out about the efficiency of a device on the component level, e.g. in a switch-mode power supply. This is often done to cross-check the efficiency you measured on a black-box level ($P_{OUT}$ vs. $P_{IN}$).
The power consumption of the device connected to the fuse: Not measurable when just the two pins of the fuse are available. You can measure the current flowing to the circuit, but you don't know the exact voltage.