Most banks will not allow you to use online bill pay with a savings account as the funding source; rather, instead it must be funded from either a checking or money market account. The reason for this is that checks can typically be written from a money market account but not from a savings account.
Update: I was having trouble wrapping my head around what the check would look like when the "pay from account" is an external bank, so I just called Bank of America and asked them. Basically, they do an ACH Withdrawal from your external bank account and route the money directly to the payee electronically. This means that your BofA account isn't touched and it won't show up on your BofA statement (but you can see it in the online bill pay history, and on your external bank's statement.) If the payee cannot be paid electronically, than you cannot use an external funding source. In other words, if a physical check is going to be sent, then it must have a BofA account as its funding source.
Even though the ACH withdrawal should technically be allowed from a savings account, I suspect that this is forbidden since the intended purpose of the ACH is actually to streamline the writing of a check.