No. Adding an asterisk to every field or almost every field on the form will do nothing but add clutter to the form. If all of your fields are required simply make a note up front saying so.
Be careful though, if you have a couple of optional fields in how you mark them. I would suggest putting '(optional)' either at the end of the field or as a placeholder in the field itself (unlike other uses for placeholders we don't care if 'optional' goes away when the user starts typing). Avoid marking optional fields with an asterisk or anything resembling a typical 'required' field.
Consider this example of a poorly designed form at UX Delante, if you had to say, on first glance, which fields were required you'd probably be wrong. The second example showing the form validation only makes matters worse.
Whatever solution you go for keep accessibility in mind. You don't want a solution that depends only on visual cues.