Are passwords updated once they have been saved?

To answer your question more directly:

No, this will not be automated, on the one hand because Bitwarden does not have any option for setting a password expiration date, and on the other hand because no password manager can automate the process of updating passwords in a way that works on every website (simply because there is no universally adopted standardization of the methods used for changing passwords on different websites, despite some recent efforts to implement at least partial standards to aid with automation).