API-650 tanks may optionally be designed for internal pressures, but it is not a requirement, and it should not be assumed that a tank is adequate for a given pressure without some basis. Generally, diesel tanks are atmospheric.
API-12F and API-12D tanks are rated for varying pressures depending on the size.
I am not aware of a requirement to pressure-test in-service tanks. Refer to API-653 for typical inspection procedures.