Your question is a bit too generic for a definitive answer, but the
answer is most likely ‘yes’.
If you are writing to a a UART then it takes many writes to the UART
registers to write the data to be transmitted character by character (of
using a FIFO, or whatever), and many reads of the registers to control
the output, and a finite time for the characters to actually be
transmitted - so you only want one task doing that at a time. If two
tasks were to access the UART at the same time you would get garbled output.