When a customer adds a beneficiary to a checking, savings, or CD account, are we required to get a new signature card?

Obtaining a new signature card is the preferred practice in making any change to access, ownership, and survivorship provisions. It clearly indicates the intent of all the parties and the date on which the change became effective. (In some cases; e.g. removal of a signatory on a checking account, the only prudent choice is to close "account A" and open "account B.")

In disputes relating to ownership, access and survivorship provisions there is often a signature card where one name was crossed out, initials were applied, little arrows were drawn or (my favorite) White Out was used. Imagine that card blown up to a 3 X 5 poster and set before the judge or jury during opposing counsel's arguments. Your bank will look foolish. Your bank will look sloppy.

At a minimum, get a new signature card.