I've recently acquired an Icom IC-745 and I'm unclear on how it's supposed
to work in CW mode. It does receive in SSB and AM, so in general the
receiver appears to work. When I press the 'mode' button to get it to CW
mode, however, it's just... silent

Joel-

As others have already mentioned, the radio might have had the CW filter
removed without replacing a jumper that was removed when the filter was first
installed.

Someone else may have mentioned the possibility of a defective diode in the
filter-selection circuit.

It is also possible that a crystal is bad in the circuit where USB/LSB/CW are
selected. Newer radios use signals derived from the Phase-Locked Loop
oscillator, but older ones had separate crystals for each mode. I don't know
where the IC-745 falls. If you have another radio to tune to the frequencies
of these crystals, you should be able to tell if one is not working. All three
should be within about 3 KHz of each other.

73, Fred, K4DII