UART pins to unpowered MCU?
It depends on the MCU, but in most cases it'll power up the MCU, and possibly the rest of the board through the MCU.  The MCU will try to run, and do odd things.  Your board will do odder things.  If your board draws enough current, it'll damage that pin on the MCU.

You need to arrange for the UART signal to stay at 0V when the MCU is off.  If the UART chip (or UART) that you are using doesn't have an enable pin (the USB UART chips that I've worked with can be configured for exactly the case you're describing), then AND the UART outputs with the microprocessor's VCC.