Is there any fire danger having 8 devices (computer & peripherals)
plugged into a single surge protector and running 24/7 under the desk
in my home office?

I don't think any Researcher can give you legal advice about possible
fire damage.  I'd kindly suggest, however, that you look into a UPS
rather than a mere surge protector for equipment that is intended to
be up 24/7. Not for fire reasons, mind you, but rather for protection
and reliability in the case of brownouts and blackouts, which surge
protectors won't protect against.

In pure practical terms, many businesses in the world run 24/7 with a
single outlet connection. Your concern may be more of a power bill
issue rather than fire hazard. As long as the standard issues: no
frayed wires, plugs fully inserted, original cabling, and no more
power draw than your outlet (15 amps, usually) can handle, I have a
sneaky feeling that you'll be safe. Otherwise, I'll have to turn off
my own 24/7 server running on a UPS at my house.

I am not a Google Answers Researcher. Still, disclaimer below applies.
If you're still worried, call your local Fire Department. And have a
fire extinguisher rated for electronic equipment nearby, just in case.
:)