Is it safe to turn it on while it's in pieces but connected to the mains?

Yes, the mains power is confined to the PSU which is has it's own safety enclosure - so it's safe.

First thing to check is the fan cable coming out from the CPU fan itself, is it plugged in?
Disconnect and re-seat.

Check any other loose cables, what components are at the other ends?

The CPU fan might actually be plugged into a socket on the motherboard intended for the grapics card fan, so don't assume that everything the BIOS is telling you is true!
If the cpu fan is working, then while it's open do check all the other fans inside. (there's often one hidden away at the bottom of the front panel.)

ONLY run the PC without a CPU fan for a few minutes at a time, enough to do diagnostics, or you'll run the risk of frying the processor.
Open the case before starting the PC to avoid unneccessary running time, the open case may also give slightly better airflow to slow down the inevitable build up of heat.

If in doubt, buy a new fan, they're not expensive and cost a darn sight less than a new CPU would cost you.