The answer to this question is dependent on a few ifs and buts. So, you will have to figure out your scenario from the description below to find the right answer.
The regular incandescent bulbs when burnt do not use electricity. This is because there is only one element in these bulbs that work i.e. the filament. If it is burnt, it means no closed loop can form for the electricity to flow. Thus, it cannot consume electricity.
CFLs and LEDs
If you are using a CFL or LED bulb and it burns, it can still consume electricity. This is because of their making. They are complex.
There is a high chance that the power supply component of these bulbs could still be active, even if the bulb seems burnt. It means there is more than one path available for the current to flow and this is where the electricity could still be flowing.
However, if the entire bulb is damaged in any way, it will not use any electricity. But figuring out the exact circuitry of the bulb and the extent of failure could be the only ways to find this out, and consequently, the electricity usage.
Would the electricity consumption be the same just like the active bulb?
The electricity usage would not be the same as that of an active bulb. Only a trace amount of electricity is used in such a scenario. So, you would not be worrying much about the electricity use if you have not removed the burnt bulb and the electric switch to it is still on the ON mode.
The type of bulb you are using, the extent of damage in circuitry, and the exact circuitry under consideration would provide an answer to this question.