While it is convenient to pick screws with magnetized tips I wonder if the magnetism can cause any damage to electronics?
You could make an old time crt monitor display funny colors, not sure about 3.5" and 5" floppies.
But unless your screwdriver is with pretty big neodymium magnet head, you are highly unlikely to damage anything. Most electronic components are not ferromagnetic and inducing strong current inside them is hard to do. You will even have hard time damaging the data on a hard drive.
Magnetic tips don't do damage unless there is power, at worst, they can change stored data.
A hard drive is sealed of by a metal case around it, and the distance to the internal components is big. So, it won't be affected. However, watch out with memory cards and ROM memory like the BIOS, although I would believe they are protected against static electricity there might still exist a small risk...
These are simple physics, the magnetic tip has a magnetic field which will induce ferromagnetic wave into the conductive materials around it. This induced power will be rather small, but depending on the amount of magnetic force the magnetic tip has it might be enough to change some bits when you get close enough to the magnetic parts of not properly shielded storage media...
It’s unlikely that the magnet in your screwdriver is powerful enough. See this question on specifics about what can be harmed.
In the old days, there used to be a lot of risk with computer innards and magnetic devices -- the hard and floppy drives used to be susceptible to magnetic scrambling.
Nowadays, not so much -- to get the density of 500 GB+ drives, the magnetic material on the platters is almost non-magnetic, and it requires intense local fields to change the bits -- such intensity is only acheivable with strong fields from the tip of the head at 0.1um or so from the surface. Nowadays, you could pick up the drive using a electromagnet from a crane and the data would not be damaged.
A magnetic screwdriver can damage certain components that operate from magnetic fields. The classical case was the old-fashioned tape recorder head (data or audio) which needed to be "degaussed" whenever you worked near it with any ferrous tools, even if not magnetized.
Of course, no one uses tape recorders any more, and for the most part things like disk drives are fully enclosed now and relatively insensitive to external magnetic fields. However, I would still advise that one not using a magnetic screwdriver when working on or around a disk drive, since drives are so thin and compact now that a screwdriver tip could get quite close to one of the heads. And, of course, diskettes can be damaged by a magnetic field that is strong enough (possibly a magnetic screwdriver laid on top of a diskette, eg). (Don't keep boot diskettes and the like in a toolbox.)
In terms of other components -- circuit boards, power supplies, CD drives, memory modules, etc -- a magnetic screwdriver (or any sort of permanent magnet less powerful than one used in an MRI machine) will not cause damage. (You can, of course, still cause damage to memory modules and the like due to static discharge, but magnetism had nothing to do with that.)
No, there's nothing to worry about, many modern laptops are full of magnets. The magnet I stick on my screwdriver when removing tricky / sticky screws came from a laptop lid. They are used to keep the lids closed on modern laptops without the need for mechanical latches (and occasionally to trigger the lid closed sensor). They are very strong magnets (much stronger than any screwdriver I've ever used) and quite often they come down very close to the hard drive, something I doubt the manufacturers would do if it caused any serious reliability issues.
We used magnetic screwdrivers in our shop...we fixed everything electronic including computers. I once saw a computer tech struggling to screw down a motherboard...I told him just use a magnetic screwdriver and he said oh no! I've read computer books that warn you never to use magnetic screwdrivers around a computer. I don't see how it could possibly harm the motherboard.