Most likely the module, if it is a separate module and not part of the SoC of the infotainment system or whatever, works over CAN bus and the car will throw errors when it doesn't detect its presence, or doesn't detect the SIM card. Might even refuse to start if that module is missing. Might be possible to remove the antenna so the car thinks it's just outside of the service area, but if it's built into the PCB and the PCB is cast into resin/silicone for waterproofing, even this might be extremely difficult. Probably the module is also serialized* so replacing it with a "dummy" module or a module from a junkyard won't spoof the system, either.
*Manufacturers have been serializing even airbags for years, making replacing a faulty one with one from a junkyard impossible.