If I use the USART of the PIC16F627 (or any other PIC with an USART)
    in async mode to communicate with a PC via a RS232 serial port, do I
    need to a transceiver chip such as the MAX232, or will it work without
    the transceiver?

If you use a hardware UART of any kind, you will need SOME sort of
transceiver circuitry.  You might be able to get away with as little as a
logic-level inverter, but the rs232 voltages levels are inverted from the
uart outputs (+5 out of the uart needs to make -3 or less, GND out of the
uart should make +3 or more.)  So you need SOMETHING.

BillW

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