Can I multiplex a thermocouple (using a MPC507) without losing significant accuracy?
Once you have the thermocouple signals in copper on the PCB, then adding another small IC, one whose legs are at the same temperature as each other, is unlikely to damage the reading accuracy significantly.

A potential problem with the MPC507 is its high RDSon, 1300ohms typical, 1800 max over temperature. Being differential, that's typically 2600ohms out and back. Some thermocouple receivers pass a small current through the thermocouple to detect open sensors. However the 31856 can handle loop resistances of 5k or 40k when detecting O/C, and its input bias current is +/-4nA maximum, which will introduce an offset of max 10uV in 2.6kohm, max 0.25C for a K type.

It looks like you'll get away with it.