I know that some monitors can get really screwed up by magnets. Are there any kinds that don’t? Also, please let me know about the actual computing hardware inside the tower or case, depending on the type of computer.
In general all TFTs are pretty much immune to magnetic fields. The same applies to SSDs.
You have to be more concerned about the cooling parts. Both, FANs for air-cooling and electrics pumps for water-cooling might be influenced by strong magnetic fields. So even if you are using an TFT and SSD you shouldn't deposit a super magnet on the case or right next to it.
As for electronics in general it depends on the strength of the magnetic field and it's distance. The best advice is to keep magnetic fields as low and as far away as possible from electronics.
There components specifically designed for use high magnetic field areas but they are probably way' more expensive.
FYI: Hard Drives are actually pretty good shielded against magnetic fields.
It usually takes a strong super magnet or degauss process to damage the Hard Drive and it's data.
In most cases you will probably kill all your computers FAN's, the BIOS chip and whatever else before the Hard Drive takes damage.
BIOS chip? Why? Is it more vulnerable than the CPU or is that just an example?
All electrical devices have an EMF (eletro-magnetic field) which is influenced by magnetic fields.
Most of them are just resistant enough for the average (low power) fields. Of course the circuits are also designed to withstand certain external influences and power fluctuations.
As for the BIOS (or computer in general) I don't know of any tests about what magnetic field strength is required to cause damage.
What you have to consider is that the BIOS mostly rests on an unshielded socket or is soldered so it will most likely be one of the first parts where checksum errors or data corruption appear.