Do "blown" fluorescent lights still use electricity?
I'm going to go out on a limb and say this question is valuable from the point of view of electronic design, as it pertains to some fundamental understanding on how fluorescent lights work.

Fluorescent lights work by accelerating electrons from the cathode to the anode in an almost-vacuum environment. In this vacuum is mercury vapour, and when the electron hits a mercury atom, that Hg atom goes into an excited state and outputs one or more photons of UV light upon decay. These UV photons then hit the phosphor-based coating on the inside of the glass tube, which converts these UV photons to visible white light.

So, in order to function, it is vitally important for these lights to have a lot of 'free' electrons available to shoot at the mercury. One way to make electrons more mobile and likely to shoot off the cathode is to heat it up, and this is what a so-called 'starter' circuit does: it is essentially nothing more than a high voltage generator and a heating coil. The heating coil heats up the electrode to mobilize the electrons and the high voltage generator (usually just a resonant LC pump) creates enough voltage for the initial 'spark' to ignite the bulb. Once electrons start flowing and the lamp is 'on', the gas inside the lamp looks  more like a plasma and is very conductive, so neither the high voltage nor the addition of heat is necessary to keep it working. Hence, it's just a starter, once the bulb is on, it is shut down.

Old-style starters would keep trying to fire the bulb even when the electrodes were entirely spent. This means that that heating coil would be running until its filament would burn out. In a lot of cases this would mean the bulb has a higher power consumption after it's died.

Modern electronic starters 'give up' after a few tries when they detect that the bulb won't start. After that they use up no or almost no energy until power is cycled to the starter.