It probably depends on the reason for the change.  Most bank resolutions provide that certain officers have authority to sign checks and take other actions on behalf of the organization.  Therefore, if the treasurer is authorized to sign and the old treasurer is replaced by a new individual, the president can change the signatory as a ministerial matter because it is in keeping with the original resolution.

On the other hand, if the president decides that it would be nice to have an assistant treasurer also empowered to sign, but that was not included within the original resolution, the board would have to agree on a new resolution to authorize the change.

It would be very unusual for a board to give the president blanket authority to determine the individuals who could sign on behalf of the organization, and it wouldn’t be a good idea.  The board is ultimately responsible for the finances and ought to keep control over the identity of the individuals who can spend the organization’s money.