Not usually, no.
However, sometimes the interest rate on your card is linked to your overall relationshipwith the bank (you get a better rate for having more and bigger accounts with them). Hopefully you're not carrying a balance, so this shouldn't be an issue for you.
It's possible the terms and conditions of your credit card agreement require you to keep another account open with the bank. Since you no longer have the T&Cs, you should call the credit card department (not the bank account department!) and ask.