If there is significant noise on the supply to the MAX31855, you will experience some errors in the temperature conversion results, so implementing a voltage regulator on VCC would be advised.
As an example, when using a stable 3.3V supply, and the MAX31855 will read about 24.4C from the internal temperature sensor of the MAX31855. With a 240kHz 40mVpp signal applied to the 3.3V supply, a sine-wave form would induce a -2.2C error, and a square-wave form would induce a +10.5C error. The supply noise impacts the internal ADC, so this will also affect thermocouple conversion accuracy. A separate, clean supply will be needed to ensure accurate temperature conversions.