Sounds like a nice a dumpster.
On the RS-232 connection, the manual says the 33120A uses DTR/DSR hardware flow control (and not the more ubiquitous RTS/CTS). Do you know if DSR is true (DB9 pin 6 > +3V)? If it is not, no characters will be sent from the 33120A (user manual pg. 198). I assume you selected RS-232 from the I/O menu via the front panel (pg. 115).
Not sure why the GPIB interface is not working. Could be something wrong with the interface. But I would first double check on the 33120A I/O config menu that you're using the right GPIB address for the "*idn?" test.
The GPIB and RS-232 are both handled by the same communications processor (U903), so there could be something wrong in that area that's causing both to fail. It talks to the main processor through an optically isolated Tx/Rx, so maybe something is flaky with the data path. It's supposed to be checked during self-test at power up.
I would try to get the RS-232 to work first before digging into anything else. It's connected directly to the comm processor, so whether it works or not could provide some clues on where to look next. Perhaps try some commands that set operating modes that you can see from the front panel. Maybe it's only broken for data coming from the unit.