Low suction pressure will lead to high compression ratio for this stage and consequence is high temperature on discharge. So trip the compressor.
High suction pressure will lead to high mass flow and hence potential overload on compressor driver. So trip the compressor.
As part of trip sequence, it is a good idea to force open all recycle valves.
In addition to high discharge temperature, a high compression ratio can exceed design limits for rod loading on some machines. If so, you must trip the compressor or risk the possibility of piston rod failure.