Permission from an interviewee to record and broadcast the interview is generally implied by the circumstances. However, subsequent uses of the interview (such as the sale of tapes containing the interview) might require permission on the basis that the interviewee’s words, or the interview as a whole, might be a “literary work” for copyright purposes. Also, the interviewee is likely to be a part owner of the copyright in the sound recording on which the interview is recorded, and therefore may be able to control reproduction and other uses of the recording.
It is common for interviewers to get written permission from interviewees to record an interview and for subsequent contemplated uses of the interview. You should clarify the uses that you want to make of the interview and ensure that the interviewee gives the necessary consents in the written document.