Magnetic field around the printer
The heated bed is a big PCB whose traces can act like a coil since heating the bed makes current flow through these traces.

By placing a metal object over it, you encounter a phenomenon called induction which will heat up the metal object (that can be the screwdriver tip) because the magnetic field generated by the "coil" of the heat bed.

Since these screwdrivers can be pretty sensitive the electrical current transferred from the heat bed to the metal bit may turn on the screwdriver LED

Also, stepper motors contain magnets and electromagnets which produce an alternating magnetic field when turned on which, by induction, can produce a little current in the spring inside the screwdriver which would act like a receiving coil thus turning on the led hooked up to it.

Try this with your printer on: put the tip of your screwdriver on your tongue (which will effectively ground it with your body) and put the handle over stepper motors or the heat bed and see if it lights brighter.