If I hold an F-1 status, can I do an unpaid or paid internship without work authorization?

Usually not. It is very hard and almost impossible to classify the work as “unpaid internship.” It is often abused by the employers who want to underpay for services or by the students who want to work “for cash” and be paid “under the table.” The U.S. Department of Labor regulations concerning unpaid internships are very strict and limiting. You can “volunteer” for a non-profit charity organization and, theoretically, do an unpaid internship which advances your knowledge and skills in the field of your study. But, if you work under the control and direction of the business and provide benefits through your work to that business, then the employer must at least pay you the federal minimum wage. From immigration standpoint, you must also be ready to demonstrate that your unpaid work does not displace potential US workers. In other words, whether you are paid or unpaid is immaterial if by virtue of providing services to the employer you are displacing potential US workers who could be hired for that job. The employer who hires an unpaid trainee without work authorization, must get no immediate benefit or advantage from the activities of the unpaid intern. The employer who is accepting the unpaid trainee without work authorization is expected to be actually burdened instead of benefiting from providing the training.